Could we have a 'soft' accepted answer for an unanswered question?
Moderators accepting answers on user's behalf after a certain time period
prove that if two real valued  functions are uniformly continuous on some interval $I$  and each is bounded on $I$ then their product $f\cdot g=f(x)\cdot g(x)$ is also uniformly continuous on $I$ . Is boundedness of each function  on $I$ is  necessary for the product How to restore the down arrow icon on the Downloads folder in the home directory? I removed the original Downloads folder in the home directory and created a new one with the same name, but the down arrow icon on the new Download folder now disappeared. Could someone please tell me how to restore that icon?   In addition, from the image above we see "Downloads" disappears also in "Places" category in the left panel. I think it should be there originally. If after restoring the down arrow icon, the "Downloads" is still not in "Places" category, I would also like to ask you to help me put "Downloads" back to "Places". Thank you.  PS: I am not using any theme. Is there a notification mechanism for when a dependency property has changed? I have two PHP scripts, both using the same session by calling session_name('MySessID').  When the first script calls the second script using curl, the second script hangs when session_start() is called.  Why would this happend? Do spice containers need to be hermetic? Spice containers purchase  I'm planning on upgrading my spice storage. Reading some advice online (e.g. , and ) I came to the conclusion that spice containers should:   Not be exposed to direct sunlight and high temperatures Be airtight   Which means that either you store it inside of some closed cupboard/wardrobe/room or you have containers which do block light and are in a relatively shady place. And the containers need to be airtight.  Available solutions  I've been browsing for some containers available online in my country and the 2 most sensible options I've seen are these:       The problem  The first one is hermetic (it has some rubber around the lid) but it's clear glass. The other one does not have the rubber, just a regular twist-off lid, but it's amber glass.  It'll be hard for me to find any space inside of cupboards to store all of my spices (I need at least 30), so I'd rather buy amber glass container, which I believe will effectively block the sunlight, even when kept on the counter. But I can't find any hermetic amber-glass containers, only ones with regular lid.  The question  Is that regular twist-off lid enough to keep spices airtight and fresh? Or should I focus on buying a hermetic container (like the one in clear glass). I could then for example paint them black to block the sunlight. I care to understand the concept behind the Borel-Cantelli Lemma better, hence I would appreciate it if someone could take the time to check if my arguments below are clear and rigorous.     Statement: Let $(X_n)_{n \geq 1}$ be a sequence of non-negative random variables that are identically distributed $X_n \sim X \geq 0 $ with $E(X)&lt; \infty$, show that $ X_n/n \to 0$ almost surely     Proof: Since $X_n \sim X$ for some non negative random variables we can easily compute that for $\epsilon &gt;0$ we have $$ \infty&gt;E(X/\epsilon) = \int_0^\infty P(X/ \epsilon &gt; t) dt = \sum_{n=1}^\infty \int_{n-1}^n P(X/ \epsilon &gt; t ) dt \geq \sum_{n=1}^{\infty} P(X/ \epsilon &gt;n) $$ Thanks to the Borel-Cantelli Lemma we obtain that $$ P ( \limsup_{n \to \infty} \lbrace X_n/n &gt; \epsilon \rbrace ) = 0 \iff P ( \liminf_{n \to \infty} \lbrace X_n/n \leq \epsilon \rbrace) = 1 $$     Question: Does the $\epsilon$ in the above depend on $k$?   I think it does and I would argue as follows $$P( \liminf_{n \to \infty} \lbrace X_n/n \leq \epsilon \rbrace )= P  \left( \bigcup_{n=0}^\infty \bigcap_{k=n}  \lbrace X_k/k \leq \epsilon (k) \rbrace \right) $$ Now using the chain of inclusions that state that we have for $\epsilon &gt;0$  $$ \lbrace \limsup |X_n-X| &lt; \epsilon \rbrace \subset \liminf \lbrace |X_n-X| &lt; \epsilon \rbrace \subset \lbrace \limsup |X_n-X| \leq \epsilon \rbrace $$ We conclude that $$ P ( \lbrace \limsup_{n \to \infty } X_n/ n \leq \epsilon(k) \rbrace)=1 $$ Where I choose the same $k \in \mathbb{N}$ as above, and then choose $k^*$ large enough such that $\epsilon(k) \leq \frac{1}{k^*}$, define the set $$ \Lambda_{k^*}:= \lbrace \limsup_{n \to \infty} X_n/n \leq 1/k^* \rbrace \implies P(\Lambda_{k^*})=1$$ and also we obtain that $\Lambda:=\bigcap_{j \geq k^*} \Lambda_{k^*}$ is still an almost sure event, i.e. $P(\Lambda)=1$ but for all realizations $\omega \in \Lambda$ we have $\limsup_{n \to \infty} X_n( \omega)/n \leq 0  \implies X_n( \omega)/n \to 0$ for all $\omega \in \Lambda$ that is$X_n/n \to 0$ almost surely $\Box$  Are my arguments given above correct and more importantly complete? Are there steps where you'd use different arguments? Infinitely many primes for quadratic residues Activate physics by keyframe I'd like to have a group of debris fall from a ceiling onto the floor, but not have them drop all at the same time. Is there a way to set a keyframe that dictates when the object enters the simulation? For example, object 1 starts falling on frame 0, object 2 on frame 5, object 3 on frame 24, etc. Regex pattern to choose data BETWEEN matching quotation marks What is the best way to detect if a user leaves a web page?  The onunload JavaScript event doesn't work every time (the HTTP request takes longer than the time required to terminate the browser).  Creating one will probably be blocked by current browsers. Finding the limit $ \lim_{x\to a^+} \frac{\cos(x)\ln(x-a)}{\ln(e^x-e^a)} $ Round a double to 2 decimal places How to show $\lim_{n \to \infty} a_n = \frac{ [x] + [2x] + [3x] + \dotsb + [nx] }{n^2} = x/2$? This question came from the prelim exam I took last month. I have a proof that seems a bit unwieldy to me (posted as an answer), so I'm opening it up to ask if there are other ways of showing this.     Let $x$ be any positive real number, and define a sequence $\{a_n\}$ by   $$ a_n = \frac{ [x] + [2x] + [3x] + \dotsb + [nx] }{n^2} $$   where $[x]$ is the largest integer less than or equal to $x$. Prove that $\displaystyle{\lim_{n \to \infty} a_n = x/2}$. Direct Link to a Comment Calculating ratio between "rows" in ArcGIS field calculator I have a problem similar to  but that is for QGIS whereas I have to perform it in ArcGIS. I have to calculate the Ratio between rows (sequentially, i.e., row1/row2, row2/row3 and so on) of a specific field based on the ID field and have to assign the results to a separate field. Here is the code and the error message:  def CalcRatio(ID,Frequency):     vDict = {}     with arcpy.da.SearchCursor(&quot;Test&quot;,['Frequency'],'ID = {}'.format(ID)) as sCur:     for Row in sCur:     vDict[Row[0]]=Row[1] # Dict[id] =Frequency     try: value = ( vDict[1] / vDict[2])     except: value = -1     return value  Message: ERROR 000539: Error running expression: CalcRatio( 1 , 50680) Traceback (most recent call last): File &quot;&lt;expression&gt;&quot;, line 1, in &lt;module&gt; File &quot;&lt;string&gt;&quot;, line 5, in CalcRatio IndexError: tuple index out of range  Here is the data OID Id  Frequency 0   1   50680 1   2   49740 2   3   48620 3   4   48300 4   5   48004 5   6   45600 6   7   44384 7   8   42720 8   9   41890 9   10  39700 10  11  38530 11  12  38207 12  13  38106 13  14  37303 14  15  37205 15  16  36527 16  17  35301 17  18  31674 18  19  28890 19  20  24612 Story about building up speed leading to increased quality of life Difference between generalized linear models & generalized linear mixed models How to check if windows has automatic updates waiting from the command line? If a Windows server (2k3/2k8) is set to "download but not install" updates, is there a way to check from the command line.. perhaps a log file or something I can check with powershell, to see if there are any updates actually waiting to be installed? I'm trying to prevent having to manually log on to each server to check, even though they want the "trigger" pulled manually. We have an automation system in place I can use (CA Autosys) - just not sure what to have it look for. How to run an executable on boot and keep it running? I have an executable build from a ndk-build program. I want to run it on a rooted Android device. How much is the possibility that Android will kill my executable?  Using adb shell, I am able to execute my executable using following commands:  adb push executable /sdcard adb shell device:/ $ su device:/ # cd /system device:/system # mount -o rw,remount /system device:/system # mv /sdcard/executable . device:/system # ./executable   My requirement is to run this executable on device boot.  I have tried following:   Write ./executable in init.rc.   On reboot init.rc reset to its original contents. I figured out that Magisk done this.  Write commands in /system/etc/init/bootstat.rc   ./executable service custom /system/executable on boot  ./system/dhandler/diag_revealer    None of the above thing is working. understanding TTNavigator