Let $(x_n)_n$ and $(y_n)_n$ be sequences in $\mathbb{R}$, such that $(x_n)_n$ converges to $x \in \mathbb{R}^+$. Suppose that $(y_n)_n$ is bounded. Prove that $$\limsup_{n \to +\infty} x_n \cdot y_n = x \cdot \limsup_{n \to +\infty} y_n$$  I know that $x=\limsup_{n \to +\infty} x_n$, so it basically comes down to proving $\limsup_{n \to +\infty} x_n \cdot y_n = \limsup_{n \to +\infty} x_n \cdot \limsup_{n \to +\infty} y_n$, but that's where I'm stuck...
Note: $x_n$ is a sequence which is not necessarily convergent.    The following was my attempt. Since $\lim_{n\to \infty}a_n=a$ then $\limsup_{n\to \infty}a_n=a$ . Also $\sup(a_nx_n)=\sup(a_n)\sup(x_n)$. Therefore, $\limsup_{n\to \infty}a_nx_n=\limsup_{n\to \infty}a_n\limsup_{n\to \infty}x_n=a\limsup_{n\to \infty}x_n$.    I am not sure if this is correct. So could someone please show me how it is done. Thanks
I have a string variable that represents the name of a custom class. Example:   string s = "Customer";   I will need to create an arraylist of customers. So, the syntax needed is:  List&lt;Customer&gt; cust = new ..    How do I convert the string s to be able to create this arraylist on runtime? Why doesn't my "Nikkor AF" lens autofocus on my Nikon D3300? I just read a description of CVE-2020-1971, which indicates that a patch was released today for openSSL, the patched version is 1.1.1i -- my version of Ubuntu 20.04.1 LTS is running openSSL version 1.1.1f.  From last March. Software Updater says that my system is up to date.  I am hoping that this openSSL update is released by Canonical by tomorrow... can anyone confirm they are working on this? This question came up after reading , in which it is re-iterated (as it has been many times on this site), that investing in the stock market, especially an index fund that tracks the top 100 / 500 companies in a country, basically always has great returns in the long run.  So my question is: Why isn't everybody rich?  If it is so easy (basically invest as heavily as you can in your youth to be rich into middle age and older), why doesn't everyone do this? There would be no need for pensions or any other savings or financial security that way, why even bother investing in assets like property that you have to manage when you can just have an index fund work for you?  I'm guessing that something is being missed here. Why is the colour picker displaying tinted colors? The colour picker is behaving not as I expect it to. I cannot move the selector towards most of the edges, setting the RGB to 1.00 for red and the rest to 0 just gets an orange colour. Also I cannot simply keep the "value" the same and select a different shade, the "value" resets.  I'm not sure if i've pressed something in error or there is a setting in the user preferences/elsewhere that needs changing. I'm using the cycles renderer. The issue occurs with any colour picking :- material, background etc. Quitting Blender and restarting does not resolve the issue. Version 2.79.6 We can prove that there is no integer $n&gt;1$ such that $2^n-1\mid 3^n-1$. This leads to the following question:     Is it true that for every pair of primes $p&lt;q$ there are only finitely many integers $n$ such that $p^n-1\mid q^n-1$? Are there two primes $p&lt;q$ and an integer $n&gt;p+q$ such that $p^n-1\mid q^n-1$?Is it true that if $n&gt;6$ then $2^n-1\nmid 5^n-1$?   Edit: Here are some examples: \begin{array}{ll} 2^{36}-1\mid 41^{36}-1, &amp;3^{12}-1\mid 97^{12}-1,\\ 5^{6}-1\mid 37^{6}-1, &amp;7^{4}-1\mid 151^{4}-1. \end{array}  Now I prove there is no integer $n&gt;1$ such that $2^n-1\mid 3^n-1.$  Proof: Denote $A=2^n-1$ and $B=3^n-1$. If $n$ is even then $3$ divides $A$ but not $B$, a contradiction.  If $n$ is odd then $A\equiv -5\pmod {12}$. Since every prime greater than $3$ is $\equiv \pm1,\pm5 \pmod {12}$, some prime factor $p$ of $A$ must be congruent to $\pm5 \pmod {12}$. As $p \mid B$, we have $3^{n+1}\equiv 3 \pmod p$. Since $n+1$ is even, we get $(\frac{3}{p})=1$, a contradiction again. Where are my installed applications? How can I set default value in ActiveRecord?  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I'm trying to get the max value for each of the past months ....  Here's the data from the query, currently sorted by date (asc):   ID      Date               Value 12410   01/03/2017 12:17    0.000178 12410   01/10/2017 11:36    0.000186 12410   01/17/2017 11:27    0.000189 12410   01/24/2017 13:09    0.000182 12410   01/31/2017 10:37    0.000169 12410   02/07/2017 11:03    0.000214 12410   02/14/2017 11:52    0.000176 12410   02/21/2017 10:51    0.000200 12410   02/28/2017 12:29    0.000194 12410   03/07/2017 08:39    0.000206   Here's the query:   select AnalysisID as "ID" , AnalysisDateTime as "Date", AnalysisValue as "Value" from AnalysisValueTbl where AnalysisID = 12410 and DatePart(m, AnalysisDateTime) = DatePart(m, DateAdd(m, -3, getdate())) and DatePart(yyyy, AnalysisDateTime) = DatePart(yyyy, DateAdd(m,-3, getdate())) or AnalysisID = 12410 and DatePart(m, AnalysisDateTime) = DatePart(m, DateAdd(m, -2, getdate())) and DatePart(yyyy, AnalysisDateTime) = DatePart(yyyy, DateAdd(m,-2, getdate())) or AnalysisID = 12410 and DatePart(m, AnalysisDateTime) = DatePart(m, DateAdd(m, -1, getdate())) and DatePart(yyyy, AnalysisDateTime) = DatePart(yyyy, DateAdd(m,-1, getdate())) order by AnalysisValue desc Around 3 months ago, I submitted a paper to a Math journal for peer-review through an online submission system. The same day I received in my e-mail an acknowledgement saying that the paper had been received.  The problem is that the status of the paper on the online submission system is still "Manuscript submitted".  I think this means that the paper has not even been assigned to an editor yet (once this happens, the editor will probably start looking for reviewers), but as far as I know submissions are assigned to editors within a few weeks (I once submitted a paper to another journal and when an editor had been assigned, the status on the online system changed to "Editor handles" or something like that).  I know that publishing is a long and complicated process and I'm not trying to rush anybody, but sometimes manuscripts are lost in the system. In fact, this has once happened to me, so I just want to make sure that the paper was not lost in the system.  Should I send an e-mail asking for the status of the paper?  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"The antipodal map of $S^2$ is the isometry $\bar m$ sending each point $P$ to its opposite point $-P$" (from textbook)  Show that antipodal points remain antipodal under any isometry of $S^2$. In other words, for each isometry f of $S^2$, if $P$ and $P'$ are antipodal, then so are $f(P)$ and $f(P')$  I am unsure how I can define the $P$ and $P'$ so that they are opposites of each other in the antipodal map, and am I supposed to apply the traditional isometries (translation, reflection and rotation) to the points? I don't understand this problem. pgAdmin 4: How to establish a connection through an SSH tunnel/interface in Windows I installed the latest pgAdmin 4 (v2.1) but I found out that (), for the moment, the only way to establish a connection through an SSH tunneling is using another outside process, an  says:  I setup a tunnel as follows: ssh -f dpage(at)hostname(dot)enterprisedb(dot)com -L 5999: hostname.enterprisedb.com:5432 -N  (...) 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