Footer rising up because of not sufficient content
How do you get the footer to stay at the bottom of a Web page?
This is problem 6.2 from the 3rd edition of Principles of Mathematical Analysis.      Problem 6.2: Suppose $f\geq 0$, f is continuous on $[a, b]$, and $\int_a^b f(x) \, dx = 0$. Prove that $f(x)=0$ for all $x \in [a, b]$.   I'm looking for a critique of my proof. It's a pretty easy problem, but I am always wary of making too bold of assumptions, especially on these low level/fundamental proofs. I'll be using Rudin's notation and refer to theorems from the text (If I should include the text of each theorem, feel free to leave a comment... I'm lazy but could probably use the TeX practice :p)  Proof: Assume, for contradiction, that $f&gt;0$. Then, for any partition $P$ we have the Lower Riemann Sum: $L(P, f)=\sum_{i=1}^n m_i \, \Delta x_i$. At least one $\Delta x_i$ must be positive, since $a &lt; b$, and each $m_i$ must be positive since we have $f&gt;0$ by assumption, so certainly $\sup f &gt; 0$. That means $L(P, f)&gt;0$. Thus, we have: $$0 = 0(b-a) &lt; L(P,f)\leq \sup L(P, f) = \inf U(P,f) =L$$ where the last string of equalities holds because our function is continuous on a compact interval, so is integrable by theorems 6.8 and 6.6. So our integral has value $L&gt;0$. This is in contradiction to our given assumption that $\int_a^b f(x) \, dx = 0$, so we must have that $f=0$ on $[a, b]$. $\Box$  So, I am wondering if my proof is correct (and is presented well). Also if someone could enlighten me as to what Rudin means when he say "Compare this with exercise 1," I'd be appreciative. Is it a hint or is there something else he expects you to notice? there are a lot of things I could compare :)...  Exercise 6.1: Suppose $\alpha$ increases on $[a, b]$, $a \leq x_0 \leq b$, $\alpha$ is continuous at $x_0$, $f(x_0)=1$, and $f(x)=0$ if $x\neq x_0$. Prove that $f$ is Riemann-Stieltjes Integrable and that $\int f \, d\alpha = 0$ Simulate INSERT key in Mac OS X Long story short, I am using Citrix to remote into my PC at work, and several applications require the use of the INSERT key, which, as everyone knows, Apple chose to eliminate from their keyboards. I've tried everything I can think of, and never been able to find an easy way to send this keycode via Citrix.  I'm sure there are other uses for this, and I've read quite a few people's questions on the subject, but no one seems to have offered a viable solution.  I have the Apple USB keyboard (full-size, with numeric keypad), and in the position where the INSERT key would normally exist, Apple placed the "fn" key. Now, I still need this key from time to time, but what I'd love to see is perhaps Control-fn or similar to send the keycode for INSERT.  If anyone knows how to achieve this, preferably without installing 3rd-party software (adware?), I'm all ears. Let $a$, $b$, and $c$ be complex numbers with $a\neq0$. Show that the solutions of $az^2+bz+c=0$ are $z_1,z_2=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, just as they are in the case when $a$, $b$, and $c$ are real numbers. Is it possible to install an earlier version of Skype on Windows 8? So when I got Steam working offline I had 26 games listed in my library but only 2 installed. Recently I brought a bunch more so I want to update my library on my Offline PC with the full list that's on my Online Laptop.  I thought that all I would need just copy the data from userdata/[ID]/config/localconfig.vdf however when I do the number goes down to 2 (the number of games I do have installed) and also the OST I got with Monster Monpiece doesn't appear. When I restore just that file the list and the music listing returns.  So what do I need to copy over from my Online Laptop to the Offline PC to update the list of games in the library? I have a population of phone calls - 200,000.  There are different reasons for each call, but lets assume the number of reasons is known. i.e. 7 different call reasons: 1) Check on order 2) Cancel order 3) Billing information 4) Account information etc...  My question is, I want to get a statistically significant sample size so the number of calls I listen to in my sample are representative of the distribution of the reasons in the overall population of 200,000 calls.  What should the sample size be? Which methods to use to calculate the sample size? When to capitalize words such as "lake", and when can the whole word be omitted? Which of the following sentences are correct?     "He gave two examples of large lakes, Lake Michigan and Lake   Superior."      "He gave two examples of large lakes, Michigan and Superior."      "He gave two examples of large Lakes, Michigan and Superior."      "He gave two examples of large lakes, Lakes Michigan and Superior."   This question is primarily about what variants are formally correct, not which one sounds the best.  I think that the question should be valid for several words of the same kind, such as mountain (Mount), figure (Figure), table (Table) and so on. Comprehensive list of activation functions in neural networks with pros/cons Are there any reference document(s) that give a comprehensive list of activation functions in neural networks along with their pros/cons (and ideally some pointers to publications where they were successful or not so successful)? Hereditary properties with nested frames (mdframed and tikz) Just looking for the difference between   2&gt;&amp;- 2&gt;/dev/null |&amp; &amp;&gt;/dev/null &gt;/dev/null 2&gt;&amp;1   and their portability with non-Bourne shells like tcsh, mksh, etc. How do you challenge the closing of one of your questions? I want that my monitor turns off when I am inactive for a while, but I do not want that I am prompted for a password when I revoke my pc. In Ubuntu 16.04 it was possible to disable the lock but still enable the screensaver as explained .  In Ubuntu 18.04 I entered the following in terminal:  dconf-editor   and disabled ubuntu-lock-on-suspend:    However, since I made this change, my monitor never suspends anymore. I tried restart but it didn't help.  Any ideas how I can disable lock but still enable suspend mode? Did any Quidditch games ever end by agreement of the Captains? The game can only end when the Snitch has been caught, or by mutual agreement of the two teams' Captains; the latter is very rare, however, as one team would have to lose. (source: HP Wikia on Golden Snitch)   Is there an example from canon of this resolution to the game ever happening? Find no. of non negative integer solutions of $a+2b+3c = 200$ I want to find number of the non negative integer solutions of $a+2b+3c = 200$ (answer can contain $a,b,c = 0$)  Example one of the solutions is $a = 100, b = 50, c = 0$  I used stars and bars trick and here i get is  $\dfrac{\dbinom{200+2}{2}}{2 \times 3}$ but it is incorrect  any type of math is welcomed    what I have learnt from stars and bars technique is to find the solutions of $x+y = n , x,y,n &gt; 0$ are positive integers but I am confused with equation of type $ax+by = n$ where $a , b&gt; 1$ What is the difference between the dashed and solid attack bars? I was flipping through my Pokemon list to see which kind of attacks they had, and I notice there was a small different between the attacks. Some attacks contained a dashed blue bar where as others contained a solid blue bar.   What is the difference and significance between the two? Value of the Hubble parameter over time There is something I don't understand about the  $H$, as it seems to clump two concepts together that I can't quite unify in my head. On the one side, we have  $$V = D H$$  which means that for a given distance $D$, there is a certain amount of new space created over time - and $H$ is simply the factor that makes this relationship work. So, for example, say we have two points 1 Mpc apart, this would mean they recede at about 70 km/s away from each other  (given our current approximation of $H$).  Now the thing I can't wrap my head around is that   $$T =\frac{1}{ H}$$  is also the age of the universe. Contrary to claims made on, say, Wikipedia this means that $H$ cannot possibly have been a constant throughout the past 13 billion years, because mathematically $1/H$ means that $H$ must be continually shrinking as the universe ages.  So if $H$ did start out as some huge value and is now shrinking over time, doesn't this mean the expansion of the universe is slowing down? Because if $H$ is shrinking, I'll get a lower value of $V$ today than I'll get tomorrow. Shouldn't the notation then be more like  $$V = DH(t)$$  So which one is it? If $1/H$ is simply the solution for $D=0$, how can we use it as the expansion-velocity-per-unit-of-distance at the same time? What's worse, how can literature say $H$ has probably been more or less constant forever and simultaneously assert that $1/H$ is the current age of the universe? What am I missing? I am looking for QGIS function to extract (or detect) edge from a raster, which was referred in here,  as answer by @MappaGnosis.  The name could have changed since then, or it may have been integrated into other tool... I cannot find it in my QGIS Processing Toolbox (2.14.10 and 2.18.2).  There are similar tools such as GRASS "i.zc Zero-crossing" and SAGA "Gaussian filter", but I am hoping to find this QGIS geoalgorithm. Let $y_n$ be a sequence such that  $\lim_{n\to \infty} y_n= y$ then prove that     $ \lim \dfrac{y_1+y_2+...+y_n}{n}= y$   My try:  As  $\lim_{n\to \infty} y_n= y$ then given $\epsilon&gt;0$ there exists $N$ such that $\forall n\ge N ;||y_n-y||&lt;\epsilon $.  Now $||\dfrac{y_1+y_2+...+y_n}{n}-y||\le n^{-1}\{||y_1-y||+||y_2-y||+...+||y_n-y||\}$  How to make this limit tend to zero? Do we have negative prime numbers?   $..., -7, -5, -3, -2, ...$ Pulling a CSV file from website into QGIS with ability to automatically refresh?