For every positive integer prove that$$n^7 -n$$ is divisible by 7  My turn : Let $$k^7 -k = 7m$$ Now i will try to prove that $$(k+1)^7 -(k+1) = 7p$$  $$(k+1)((k+1)^6 -1)$$ but i stopped here !  What should i do ?
I have to prove this : "$n^7-n$ is a multiple of 7". This is what I have done this so far :  $P(n):n^7-n$ On putting $n=1,$ $P(1):1^7-1=0$, which is a multiple of 7. So, $P(1)$ is true. Let $P(k)$ be a multiple of 7. So, $k^7-k$ is a multiple of 7. So, $k^7-k=7m$, where $m$ is any natural number. Let $P(k+1)$ be true, Hence, $(k+1)^7-(k+1)$ is a multiple of 7.  Now, how do I evaluate $(k+1)^7$ without using the binomial theorem ? Or is there some other way to do it ?
If $(X_n)$ is i.i.d. and $ \frac1n\sum\limits_{k=1}^{n} {X_k}\to Y$ almost surely then $X_1$ is integrable (converse of SLLN) Let $(\Omega,\mathcal F,P)$ be a finite measure space.  Let $X_n:\Omega \rightarrow \mathbb R$ be a sequence of iid r.v's  I need to prove that if:  $ n^{-1}\sum _{k=1}^{n} {X_k} $ converges almost surely to $Y$ then all $X_k$ have expectation.  If I understand correctly then $X_k$ has expectations means $X_k$ is in $\mathcal L^1(\Omega)$.  And I know that on finite measure space Converging in expectations is converging in $\mathcal L^1(\Omega)$ and it's stronger than almost sure convergence.  And I know that from linearity of expectation even if one of the sequence is not in $\mathcal L^1(\Omega)$ then $Y$ is not in $\mathcal L^1(\Omega)$.  How do I continue? Can't indent heredoc to match code block's indentation If there's a "First World Problems" for scripting, this would be it.   I have the following code in a script I'm updating:  if [ $diffLines -eq 1 ]; then         dateLastChanged=$(stat --format '%y' /.bbdata | awk '{print $1" "$2}' | sed 's/\.[0-9]*//g')          mailx -r "Systems and Operations &lt;sysadmin@[redacted].edu&gt;" -s "Warning Stale BB Data" jadavis6@[redacted].edu &lt;&lt;EOI         Last Change: $dateLastChanged          This is an automated warning of stale data for the UNC-G Blackboard Snapshot process. EOI  else         echo "$diffLines have changed" fi   The script sends email without issues, but the mailx command is nested within an if statement so I appear to be left with two choices:    Put EOI on a new line and break indentation patterns or Keep with indentation but use something like an echo statement to get mailx to suck up my email.   I'm open to alternatives to heredoc, but if there's a way to get around this it's my preferred syntax. I've tried to setup an autologin, but that isn't possible, because my home is encrypted. What are the reasons for this?  PS. Here I have described my problem: . Disk Utility cannot erase USB drive (error -69877: couldn't open device) Disk Utility (macOS Catalina 10.15.2) fails to erase my USB drive (64GB key, currently formatted in exFAT).  /dev/disk2 (external, physical):    #:                       TYPE NAME                    SIZE       IDENTIFIER    0:      GUID_partition_scheme                        *64.1 GB    disk2    1:                        EFI EFI                     209.7 MB   disk2s1    2:       Microsoft Basic Data Lexar                   63.9 GB    disk2s2   I've tried a bunch of commands that I've found here and there in the terminal, but none of them worked, like:  ➜  ~ diskutil eraseVolume exFAT MyName /dev/disk2 Started erase on disk2 Unmounting disk Error: -69888: Couldn't unmount disk   ➜  ~ diskutil eraseVolume exFAT MyName /dev/disk2s2     Started erase on disk2s2 Lexar Unmounting disk Erasing Error: -69760: Unable to write to the last block of the device   ➜  ~ diskutil eraseDisk FAT32 MYSD MBRFormat /dev/disk2 Started erase on disk2 Unmounting disk Creating the partition map Error: -69877: Couldn't open device   ➜  ~ diskutil eraseDisk free EMPTY /dev/disk2 Started erase on disk2 Unmounting disk Creating the partition map Error: -69877: Couldn't open device   My USB drive appears as read only when i do CMD+I, I tried repairing the permissions but that failed too.  ➜  ~ diskutil resetUserPermissions /dev/disk2 `id -u` Ready to reset user permissions on disk2 Error encountered attempting to reset permissions for user 501 home directory on disk2: Permissions are not enabled on the disk (-69861)   Thx in advance for the help :) Can "but also" be split in "not only ... but also"? I think I generally agree with the statement  When using not only . . . but also in a sentence, parallelism should be the goal. It means that the words following both parts of this correlative conjunction (i.e., not only and but also) should belong to the same parts of speech.   However, the Cambridge dictionary has an example that is not 100% parallel in my eyes, as &quot;but also&quot; is split:  Not only did she forget my birthday, but she also didn’t even apologise for forgetting it.   What I would expect, with maximum parallelism, is  Not only did she forget my birthday, but also she didn’t even apologise for forgetting it.  Is my expectation wrong? Is my formulation acceptable? Why is it usually "friend of his", but no possessive apostrophe with "friend of Peter"? As  shows, we nearly always use the possessive form of personal pronouns for friend of mine/his/ours/etc.  But when it comes to actual names, we prefer  without the possessive apostrophe. That preference is even more marked with, say, . Not that I think the usage itself is particularly American - it's much the same with .  Personally, I find friend of him grates. In general I've no strong feelings either way as to whether it's friend of Peter or friend of Peter's (though I deplore the possessive in ), but in line with many others, I really don't like the possessive in relation to things like countries.  Why is this?   Edit: Noting an apparent "progression" (pronoun -> person -> nation) marked by reduction in use of the possessive, I checked at a finer "granularity". NGram shows that although it does occur,  virtually "flatlines" against friend of mine. But the bias reduces through of you, of us, and by the time I get to  it's much less extreme. There seems to be something "egocentric" about the double possessive.  Presumably when babies learn to speak, they soon notice that possessive pronouns, possessive apostrophes, and the word "of", all do the same job. Parents would correct a child who says "of mine's", but probably wouldn't even notice the same "redundancy" in "of Peter's". Younger speakers are unlikely to even be talking about something "of America's". Perhaps as we mature we tend to discard the "double possessive" for the more "distant" things that only adults are likely talk about, but we keep it for "closer" people because that's how we spoke when we were younger.  EDIT2 I note that I'm a great fan of him is vanishingly rare compared to ...fan of his, but with ...fan (of John) the double possessive occurs far less often than ...friend (of John's). Usage seems to be affected by the noun before "of" as well as the one after it. This is getting complicated... I'm new to Ubuntu, it looks like it is better than Windows. Last day I tried to install Kget (the download manager) but during the process my computer suddenly stopped working. So I had to switch it off. After I logged into Ubuntu again I can't install any program. The Ubuntu Software Center says:  Items cannot be installed or removed until package catalog is repaired. Do you want to repair it now?   After I clicked repair it says (after few seconds):  Package operation fails - The installation or removal of a software package failed   I've tried several times, but the same problem occurred. Can anyone tell me a solution for this problem? Copy the contents of a file into the clipboard without displaying its contents How to copy the contents of a file in UNIX without displaying the file contents. I don't want to cat or vi to see the contents.  I want to copy them to clipboard so that I can paste it back on my windows notepad.  I can't copy the file from that server to another due to access restrictions. How would you evaluate $\liminf\limits_{n\to\infty} \ n \,|\mathopen{}\sin n|$ How would you evaluate the limit inferior of the sequence $n\,|\mathopen{}\sin n|$? That is, $$\liminf\limits_{n\to\infty} \ n \,|\mathopen{}\sin n|$$    Edit. Let $\mu$ be the irrationality measure of $\pi$. Since $\mu$ is not known, I will split the question:   Assuming $\mu &gt; 2$, what is $\liminf\limits_{n\to\infty} \ n\,|\mathopen{}\sin n|$? Assuming $\mu = 2$, what is $\liminf\limits_{n\to\infty} \ n\,|\mathopen{}\sin n|$?     I kind of like its graph... What is the difference between const_iterator and non-const iterator in the C++ STL? What is the difference between a const_iterator and an iterator and where would you use one over the other? Can't install wine from winehq.org on Ubuntu (actually Lubuntu) 18.04 LTS I have followed all the instructions on .  In particular, I have run the following commands: sudo dpkg --add-architecture i386  wget -nc https://dl.winehq.org/wine-builds/winehq.key sudo apt-key add winehq.key sudo apt-add-repository 'deb https://dl.winehq.org/wine-builds/ubuntu/ bionic main'   The first three of the above seemed to work fine.  However, the last command gives me the following error: Some packages could not be installed. This may mean that you have requested an impossible situation or if you are using the unstable distribution that some required packages have not yet been created or been moved out of Incoming. The following information may help to resolve the situation:  The following packages have unmet dependencies:  winehq-stable : Depends: wine-stable (= 5.0.0~bionic) E: Unable to correct problems, you have held broken packages.  I'm running Lubuntu 18.04.3 on a USB live stick.  (However, if I remember correctly, a couple days ago I got the exact same error on a normal hard disk Lubuntu 18.04.3 installation.) I've read that a lot of people get similar errors.  Unfortunately I haven't found any solutions online that worked. Any help would be much appreciated.  Thanks! EDIT:  Thanks for the comments and suggestions everyone.  I'll try them out later today. In the mean time, here's something I don't understand.  Ubuntu 18.04 has been around for almost two years now.  In all that time, surely somebody has successfully installed wine, right?  What did they do to get it working? I have two limit problems which are quite similar, so I've put them both into this post.   $((4^{10}+2^{n})^{\frac{1}{n}})$ $((3n^{2}+n)^{\frac{1}{n}})$   Attempt:  For 1, I'm not sure if this is permissible. I know that $\lim_{n \to \infty}(x^{n}+y^{n})^{\frac{1}{n}} = \max\{x,y\}$. Obviously $4^{10}$ is a constant so it's not quite in the same form, but I believe it is still okay to use this result, and so the limit is $2$.  For 2, I've got $((3n^{2}+n)^{\frac{1}{n}}) = (n(3n+1)^{\frac{1}{n}})$, but then I'm not too sure how to proceed.   I also thought of taking the log and proceeding that way, however in the notes I'm learning from they haven't covered l'hopital yet, so I'm trying to not use it.  Thanks. $K\subseteq \mathbb{R}^n$ is a compact space iff every continuous function in $K$ is bounded. I need to prove that $K\subseteq \mathbb{R}^n$ is a compact space iff every continuous function  in $K$ is bounded.  One direction is obvious because of Weierstrass theorem. How can i prove the other direction? I tried to assume the opposite but it didn't work for me.  Thanks a lot. How to use a variable for a key in a JavaScript object literal? I want to add additional items to the data returned as . So for every function/query that returns a WP_Post object I want my additional data added.  Example of returned result:  WP_Post (object) =&gt; [     // Default data returned by WP_Post     ID                  =&gt; int,     post_author         =&gt; string,     post_name           =&gt; string,     post_type           =&gt; string,     post_title          =&gt; string,     post_date           =&gt; string,     post_date_gmt       =&gt; string,     post_content        =&gt; string,     post_excerpt        =&gt; string,     post_status         =&gt; string,     comment_status      =&gt; string,     ping_status         =&gt; string,     post_password       =&gt; string,     post_parent         =&gt; int,     post_modified       =&gt; string,     post_modified_gmt   =&gt; string,     comment_count       =&gt; string,     menu_order          =&gt; string,      // Additional data I want to add     extra_data_1        =&gt; array,     more_data_key       =&gt; string,     another_added       =&gt; string ]   For example when the functions  or  are run they will return the WP_Post object along with my additional data.  I've tried finding the appropriate  but this has been unsuccessful.  I am hoping I can do something like:  // This is concept code add_action('pre_wp_post_return', function($data) {     $data-&gt;extra_data_1     = get_post_meta($data-&gt;ID, 'extra_data');     $data-&gt;more_data_key    = get_post_meta($data-&gt;ID, 'more_data', true);     $data-&gt;another_added    = get_post_meta($data-&gt;ID, 'another_data', true);      return $data; });   My reasoning is I am having to build a custom API for WP that uses a range of different core functions which return WP_Post objects. Being able to add my additional data in one place would prevent me from duplicating code.  I hope this is clear enough. Any help would be great! "Holmby hall" or "The Holmby Hall", does the hall have to be capital in either of them? We have $n$ charged and $n$ uncharged batteries and a radio which needs two charged batteries to work. How to run a script without closing the terminal? I have a script in file bla.sh and it is executable. When I click on it, the script is executed and the window is closed. I'd like the window to stay open.  Something like command cmd /k** command in Windows.  P.S. I don't want to use pause, but I want to able to write more commands after the script was executed. I was looking for a tree or graph data structure in C# but I guess there isn't one provided.  explains a bit about why.  Is there a convenient library which is commonly used to provide this functionality?  Perhaps through a strategy pattern to solve the issues presented in the article.  I feel a bit silly implementing my own tree, just as I would implementing my own ArrayList.     I just want a generic tree which can be unbalanced.  Think of a directory tree.  C5 looks nifty, but their tree structures seem to be implemented as balanced red-black trees better suited to search than representing a hierarchy of nodes. Define a linear operator $T:C[0,1] \to C[0,1]$ as follows: $$Tf(x) = f(x) + \int_0^x f(u)du$$ It is easy to show that $T$ is a bounded linear operator. The statement also (1) claims that $T$ is one-to-one and onto, and (2) asks to compute $T^{-1}$. (Hence $T$ is an isomorphism due to a corollary of Open Mapping Theorem)  I need help to prove $T$ is onto: given any $g \in C[0,1]$, how to decompose it into a sum between a continuous $f$ and its antiderivative. I have trouble decomposing $g(x)=|x|$, not to mention bizarre functions like Weierstrass function. It seems impossible.  Thanks in advance for help!