Let $B$ be the set of all the upper bounds of the non-empty bounded subset $A\subseteq\Bbb R$. Prove that $\inf B=\sup A$.   I divided it into two areas ($\inf B&gt;\sup A$, $\inf B&lt;\sup A$) and tried to show a contradiction. Is it the right approach? How can I prove it?
Let $A$ be a nonempty and bounded below, and define $B= \{b\in \mathbb{R}: b$ is a lower bound for $A\}$. Show sup$B$ = inf$A$.   So far I have: Let $A$ be nonempty and bounded below. This implies $\exists l \in A, \forall a \in A, l\leq a$ and $l =$ inf$A$. Let $B = \{b\in \mathbb{R}: b$ is a lower bound for $A\}$ and let $M=$ sup$B$. This implies $\exists b\in B, \forall b \in B, b\leq M$. By definition of $B$, we know inf$A \in B$. Since $b\leq$ sup$B$, with sup$B$ the largest element in $B$, we have inf$A \leq M =$ sup$B$, so inf$A \leq$ sup$B$.  I realize I have to prove that sup$B \leq$ inf$A$ to get that sup$B$=inf$A$. I am having trouble doing so.   Part B: Use (a) to explain why there is no need to assert that greatest lower bounds exist as part of the Axiom of Completeness.
What is the difference between weight and mass? How to change the theme of Snap Apps? I am enjoying the new "snap" applications, but apps installed using this method doesn't seem to be using the system gtk theme I have set.   Some use Adwaita or worse, Raleigh.   How can I have them use the standard Ambience theme I'm using across the whole system? I am trying to run the following model:         I(week^2):mutation_status +         (week + I(week^2) | subject_id) ,          data = sim_dat)  This is the output I get from this model. The correlation between week and I(week)^2 is rather high (-0.95) and I was curious to know how I can change my data simulation code to lower that value. I am also having issues with convergence and would love to get thoughts on how to avoid issues regarding convergence. Thank you! Linear mixed model fit by REML. t-tests use Satterthwaite's method [lmerModLmerTest]  Formula: fetal_weight ~ week + mutation_status + week:mutation_status + I(week^2) +  I(week^2):mutation_status + (week + I(week^2) | subject_id) Data: sim_dat  REML criterion at convergence: 114  Scaled residuals:      Min      1Q  Median      3Q     Max  -1.9343 -0.3980 -0.0075  0.4168  1.9928   Random effects:  Groups     Name        Variance  Std.Dev. Corr         subject_id (Intercept)   1.51524  1.2309                          week         61.01832  7.8114   0.16                   I(week^2)   496.20236 22.2756  -0.19 -0.95  Residual                 0.05892  0.2427              Number of obs: 100, groups:  subject_id, 20  Fixed effects:                            Estimate Std. Error       df t value       Pr(&gt;|t|)     (Intercept)                  3.1331     0.4226  17.9961   7.413 0.000000713699 *** week                        -1.2465     3.5210  18.0053  -0.354          0.727     mutation_statusY             0.5061     0.5977  17.9961   0.847          0.408     I(week^2)                   34.5706    10.8146  18.0084   3.197          0.005 **  week:mutation_statusY       -1.0697     4.9795  18.0053  -0.215          0.832     mutation_statusY:I(week^2) 202.5521    15.2942  18.0084  13.244 0.000000000101 *** --- Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1  Correlation of Fixed Effects:             (Intr) week   mttn_Y I(w^2) wk:m_Y week        -0.160                             muttn_sttsY -0.707  0.113                      I(week^2)    0.145 -0.966 -0.103               wk:mttn_stY  0.113 -0.707 -0.160  0.683        mtt_Y:I(^2) -0.103  0.683  0.145 -0.707 -0.966  Code for data simulation: set.seed(123)  # set the seed J &lt;- 20  # number of individuals (clusters) cs &lt;- 5  # number of time points (cluster size) gam &lt;- c(0, 0.75, 1.34)  # fixed effects G &lt;- matrix(c(1.75, 0, 0,               0, 0.0125, 0,               0,0,0.0625),              nrow = 3)  # random effect variances (G-matrix) sigma2 &lt;- 0.08  mutation_status &lt;- rep(0:1, each = 5) mutation_status &lt;- rep(mutation_status, times = 10)  X &lt;- cbind(1, seq_len(cs),             (seq_len(cs))^2)  # for each individual X &lt;- X[rep(seq_len(cs), J), ]  X &lt;- X[rep(seq_len(cs), J), ]  X[,2] &lt;- X[,2]*0.05 X[,3] &lt;- X[,3]*0.05  # repeat each row cs times pid &lt;- seq_len(J)  # individual id pid &lt;- rep(pid, each = cs)   # Generate person-level (lv-2) random effects uj &lt;- lmf::rmnorm(J, mean = rep(0, 2), varcov = G)  # Generate repeated-measure-level (lv-1) error term eij &lt;- rnorm(J * cs, sd = sqrt(sigma2))  # Compute beta_j's betaj &lt;- matrix(gam, nrow = J, ncol = 3, byrow = TRUE) + uj  # Compute outcome: y &lt;- rowSums(X * betaj[pid, ]) + eij + mutation_status*X[ , 3]*10 + 3  # Form a data frame sim_dat &lt;- tibble(y, time = X[ , 2], pid, mutation_status) sim_dat &lt;- sim_dat[,c(3,2,4,1)] colnames(sim_dat) &lt;- c(&quot;subject_id&quot;, &quot;week&quot;, &quot;mutation_status&quot;, &quot;fetal_weight&quot;) sim_dat$week_2 &lt;-sim_dat$week*5 + 15  sim_dat$mutation_status &lt;- as.factor(ifelse(sim_dat$mutation_status == 0, &quot;N&quot;, &quot;Y&quot;)) sim_dat$subject_id &lt;- as.factor(sim_dat$subject_id ) #Model Building No indent in the first paragraph in a section? I am just curious that the following format looks good to you or not? The preamble I used is just   \documentclass[12pt,a4paper]{article}  \usepackage{amsmath} \usepackage{amsthm} \usepackage{amssymb}   I don't feel it's so good because the first paragraph has no indent but the second does. If the first paragraph is much longer, than it's fine. But if not, it looks strange. Should I leave it, or you have suggestions for this? Prove that if $a$ and $b$ are relatively prime, then $\gcd(a+b, a-b) = 1$ or $2$ Prove that if $a$ and $b$ are relatively prime, then $\gcd(a+b, a-b) = 1$ or $2$.  I started off by putting $\gcd(a+b, a-b) = d$. This implies that there are two relatively prime integers $x_1, x_2$, such that   $dx_1 = a+b$   $dx_2 = a -b$  Adding the first equation to the second gives us: $d(x_1 + x_2) = 2a$, and subtracting the second from the first gives us $d(x_1 - x_2) = 2b$. This implies that $d\mid2a, d\mid2b \implies \gcd(2a, 2b) = d \implies d = 2\cdot \gcd(a,b) \implies d = 2$.   This is based on the fact that $\gcd(x_1 + x_2, x_1 - x_2) = 1$, which is in fact the statement of the problem. How do I prove this? Can I somehow use the Euclidean Algorithm for this. Suppose $n$ is a non negative integer $\geq 4$ and $\sigma\in S_n$ a permutation. Conjugacy classes in $S_n$ are completley determined by the cycle structure of $\sigma$. If we let the alternating group $A_n$.act on $S_n$ by conjugation, the orbits coincide with the conjugacy classes in $S_n$ except for those (even) permutations whose cycle structure is composed of even cycles (which corresponds to cycles of uneven length) of distinct order (fixed points are treated as cycles of length $1$.) See   For instance a $3$-cycle in $A_4$ has cycle structure $[3,1]$ which fits the bill, and a $5$-cycle in $A_5$ or $A_6$ works too, since the cycle structure is $[5]$ and $[5,1]$ respectively. On the other hand, a $3$-cycle in $A_5$ will does not satisfy the criteria as it has cycle structure $[3,1,1]$ with two fixed points.  In this case, the conjugacy class of $\sigma$ inside $S_n$ splits in $2$ orbits of the conjugation action of $A_n$, and if $\tau$ is any odd permutation, for instance $(12)$, then we have $$\mathrm{Conj}_{S_n}(\sigma)=\mathcal{O}(\sigma)\coprod\mathcal{O}(\tau\sigma\tau^{-1})$$  My question is How can we differentiate between the two orbits? Does it have geometric meaning?    I can see that this information can be useful when looking at representations of the alternating groups, since there are as many irreducible representations as conjugacy classes. As for some geometric interpretation, I ask because there is one for the three cycles in $A_4$. If $S_4$ is understood as the group of symmetries of a regular tetrahedron, then $A_4$ is the subgroup of direct symmetries, and a $3$-cycle corresponds to a rotation of angle $\frac{2\pi}{3}$ or $\frac{4\pi}{3}$ with axis passing through one of the vertices. Conjugacy classes inside $A_n$ preserve the angle, and that's how we can tell them apart. In this case the computations are so simple we don't really need this picture, but I think it is a nice way to understand the fact $3$-cycles in are not all conjugate inside $A_4$... Is "do something by date X" inclusive? Did English ever have a formal version of "you"? I need help to find the Galois Group of $x^5 +1$. I know that it has a 5-cycle and a 4 cycle and is not a subgroup of $A_5$. Thanks! Is there such a thing as charge density in quantum mechanics? Is there such a thing as a time-varying charge density in quantum mechanics? I am looking at the tungsten filament in a light bulb. Am I allowed to say that there is a wave function which describes that filament, and that in some sense there is a time-varying charge density (which some people call a probability) associated with the square of that wave function? There are electrons which have a probability distribution, and there are nuclei which are vibrating within the lattice, but all in all is there, or is there not, a time-varying charge distribution associated with that glowing filament? And if not, why not?  EDIT: There seems to be some question as to just what kind of charge-density fluctuations I am talking about. Let me clarify: in a hydrogen atom in the ground state, I would say there are NO charge density fluctuations. But if there is a superposition of the ground state and a 2p state, I would say the charge distribution fluctuates sinusoidally at the difference frequency of the two states. THAT is the kind of charge density fluctuation I am asking about for in the tungsten filament. Does it or doesnt' it?  I hope this clarifies the intent of my question.   RE-EDIT: In a separate (and rather hard-fought) discussion, it appears that people who know more than me agree that there is indeed an oscillating charge distribution in the hydrogen superposition as described above (see  ).  So if there is an oscillating charge in the hydrogen atom, why wouldn't there be oscillating charge in a heated tungsten filament? In Italian, when I write a sentence about an action (e.g. eating fish, playing), I would use the infinitive, such as  mangiare pesce fa bene alla salute (which literally is "to eat fish is healthy"), or giocare aiuta ad imparare (which literally is "to play helps to learn").   I would use the gerund in giocando si impara, but that would roughly mean "When playing, you learn." and in this case the infinitive would not be correct. (Giocare si impara is not what you would say in Italian.)  Should I use the infinitive, or the gerund, in sentences similar to the following ones?   To play with fire is dangerous. To win the lottery is just a matter of luck. To escape from Alcatraz is impossible. To practice makes you perfect.   Is the infinitive always wrong, when the verb is the subject of a sentence? Export cycles material to unity I have been experiencing an issue that once I have a material created and looking the way I want it, I cannot seem to export it correctly with the textures mixed.  I don't know the correct words to use, but in blender I have the material created using the nodes editor, and I have placed two textures inside of there (one for the rock texture and one for moss, my object is a rock). When I export this model and import to Unity (the game engine I am using), I find a materials folder but the material inside of it does not have the textures assigned, it's just a plain white material?  How can I export my material from blender in a way that it will allow me to import it WITH TEXTURES into Unity? Bash is not finding a program even though it's on my path I have a program on my path.  The program runs when executed with a full path specified.  But the program cannot be found when I run it with just its name.    Essentially, I want to understand how the below output is possible, and how to fix it so that my program can actually be found without a full path specified:  root:/usr/local/bin# ./siege **************************************************** siege: could not open /usr/local/bin/etc/siegerc run 'siege.config' to generate a new .siegerc file **************************************************** root:/usr/local/bin# echo $PATH /usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin:/usr/games root:/usr/local/bin# siege bash: /usr/bin/siege: No such file or directory root:/usr/local/bin# wtf!?!?   I'm on Ubuntu 12.04 using bash.  Also please note the warning output from siege is not relevant for the purposes of this question, as I am only interested in whether or not the program can be found and invoked. Nested regression model Is there a sorted collection type in .NET? I'm looking for a container that keeps all its items in order.  I looked at SortedList, but that requires a separate key, and does not allow duplicate keys.  I could also just use an unsorted container and explicitly sort it after each insert.  Usage:   Occasional insert Frequent traversal in order Ideally not working with keys separate from the actual object, using a compare function to sort. Stable sorting for equivalent objects is desired, but not required.  Random access is not required.   I realize I can just build myself a balanced tree structure, I was just wondering if the framework already contains such a beast. Destroying a spammer does not delete their comments I am learning Linux and I was trying the gzip command. I tried it on a folder which has a hierarchy like  Personal/Folder1/file1.amr Personal/Folder2/file2.amr Personal/Folder3/file3.amr Personal/Folder4/file4.amr   I ran     "gzip -r Personal" and now its like  Personal/Folder1/file1.amr.gz Personal/Folder2/file2.amr.gz Personal/Folder3/file3.amr.gz Personal/Folder4/file4.amr.gz   How do I go back? I have been looking at this for hours and it isn't making anymore sense than it did in the first hour.   If $a$ und ${x_{0}}$ are positive real numbers and ${x_{k}}$ defined as follows, prove that ${x_{k}}$ is monotone decreasing and bounded, then calculate the limit.  ${x_{k}} = \frac{1}{2}\left({x_{k-1}+\frac{a}{{x_{k-1}}}}\right)$  What I though I had to do was pick an ${x_{0}}$ and solve for ${x_{k}}$, so I picked ${x_{0}}$. Then I wanted to put the result back into the function to get ${x_{k+1}}$, which I still believe is what I'm supposed to be doing, but I don't understand what I am getting as a result.  I get that I should prove it is decreasing, then that it is bounded,  then address the limit, but the how is missing. windows bat file error What does a Terminator do after its mission is accomplished? A Terminator is essentially given a mission or an objective ("Go back in time and kill Sarah Connor" or something of the like).  What does a Terminator do once its mission is accomplished?  We see in Terminator 2: Judgment Day (the live-action movie) that the  (played by Arnold Terminator Schwarzenegger) completed its mission of protecting the Connors, and then sort-of just melted itself into nothingness (presumably to protect the world from discovering Terminators again).  But what would the  or the  (in Terminator 3) have done?   Are they given multiple directives ("Go kill Sarah Connor, then go kill Robert Baratheon, and then have a nice hot bath; you've earned it")?  This question is related, but not a dupe: