I've got this task to prove that, given $\forall x \geq a$ :  f is monotonically decreasing,  $ f \geq {0}$  $\int_a^\infty f(x)dx $ converges.  Then $\lim\limits_{x\to\infty}xf(x) = 0 $  I proved successfully that $\lim\limits_{x\to\infty}f(x) = 0$ but I don't know if it helps me anyhow.
Let $f(x)$ be a monotone non-increasing function such that $\int_{0}^{\infty}f(x) dx$ converges. Prove: $\lim\limits_{x \to \infty} xf(x)=0$. My question is, why can't I simply contradict any other possibility by using the Integral Limit Comparison Test with $1\over x$? I am, after all, to show that $f(x)=o(x)$ as $x\to \infty$. I really don't understand why monotony is crucial here. I could use some help.
Does existence of $\gcd$ implies PID? We know that if $A$ is a PID, then we can guarantee the existence of a $\gcd$ between two elements. My question is, does the converse hold? If I know that every two elements have a $\gcd$, do I know that the ring is a PID? My intuition says no, in that case I would like a counter-example.  Thanks. In the recently launched trailer of Fantastic Beasts: The Crimes of Grindelwald, there are people Apparating inside Hogwarts grounds.                                  As these people later question Professor Dumbledore, it seemed they were Ministry officials.  Have we ever encountered such Apparition before? Or, was this anti-Apparation charm put into place in recent times? Infinite Cartesian product of countable sets is uncountable Let $\{E_n\}_{n\in\mathbb{N}}$ be a sequence of countable sets and let $S=E_1\times\cdots\times E_n\times\cdots $. Show $S$ is uncountable. Prove that the same statement holds if each $E_n=\{0,1\}$.  By the definition of Cartesian product of sets,   $$\displaystyle S=\Pi_{n\in\mathbb{N}} \{f\colon\mathbb{N}\rightarrow\cup_{n\in\mathbb{N}}E_n\mid\forall n, f(n)\in E_n\}$$  If $E_n=\{ 0,1\}$, then  $$\displaystyle S_{01}=\Pi_{n\in\mathbb{N}}\{0,1\}=E^{\mathbb{N}}$$, where $E=\{0,1\}$.  By a theorem, $\cup_{n\in\mathbb{N}} E_n$ is countable since the sequence is countable.  I'm not sure how to go on from here to show $S$ is uncountable. Can we say anything about the function $f$ that maps a countable $\mathbb{N}$ to another countable union of sequence of countable sets? Microsoft Exchange 2013 is not receiving internet email Is “ ’s ” ever correct for pluralization? Determine the set of points $z$ that satisfy the condition $|2z|>|1+z^2|$ Determine the set of points $z$ that satisfy the condition $|2z|&gt;|1+z^2|$  I tried to redo this problem and got to this point  $|2z|&gt;|1+z^2|$ $\Rightarrow$ $2|z|&gt;1+|z^2|$ $\Rightarrow$ $2|z|&gt;1+z\overline z$  Let $z=x+iy$ then   $$2\sqrt{x^2+y^2}&gt;1+(x+iy)(x-iy)$$  $$2\sqrt{x^2+y^2}&gt;1+(x^2+y^2)$$  $$0&gt;1-2\sqrt{x^2+y^2}+(x^2+y^2)$$  $$0&gt;(1-\sqrt{x^2+y^2})^2$$  since $x,y$ are real number , so there is no solution for this inequality? How do I pass a function as a parameter to in elisp? I'm trying to pass one method to another in elisp, and then have that method execute it.  Here is an example:  (defun t1 ()   "t1")  (defun t2 ()   "t1")  (defun call-t (t)   ; how do I execute "t"?   (t))  ; How do I pass in method reference? (call-t 't1) How do I unload (reload) a Python module? I have a long-running Python server and would like to be able to upgrade a service without restarting the server. What's the best way do do this?  if foo.py has changed:     unimport foo  &lt;-- How do I do this?     import foo     myfoo = foo.Foo() On the position of multiple sub and superindices How to rename file as below  AS1100801000002.RAW7AGS  AS1100801001008.RAW7AH4 AS1100801002001.RAW7AH9 AS1100801003002.RAW7AHE AS1100801004009.RAW7AHT AS1100801005002.RAW7AHY AS1100801010002.RAW7AJ3   to new name  AS1.txt AS2.txt AS3.txt AS4.txt AS5.txt AS6.txt AS7.txt Recover accidentally removed files on NTFS partition I accidentally removed some files on a NTFS partition  How can I recover them staying inside Linux without rebooting ? Proving that $7^n(3n+1)-1$ is divisible by 9 I'm trying to prove the above result for all $n\geq1$ but after substituting in the inductive hypothesis, I end up with a result that is not quite obviously divisible by 9.   Usually with these divisibility induction problems, it falls apart nicely and we can easily factorise say a 9 if the question required us to prove that the expression is divisible by 9. However in this case, I do not end up with such a thing.  My work so far below:  Inductive Hypothesis: $7^k(3k+1)-1=9N$ where $N\in\mathbb{N}$  Inductive Step:  $7^{k+1}(3k+4)-1 \\ =7\times 7^k(3k+1+3)-1 \\ =7\times \left [ 7^k(3k+1)+3\times 7^k \right ] -1 \\ = 7 \times \left [ 9N+1 + 3 \times 7^k \right ] -1 \\ = 63N+21\times 7^k+6 \\ = 3 \left [ 21N+7^{k+1}+2 \right ]$   So now I need to somehow prove that $21N+7^{k+1}+2$ is divisible by 3, but I'm not quite sure how to proceed from here... Disable back button in android How do I use multirow in LaTeX? How to install League of Legends in Ubuntu How to calculate the summation $(\sum_{p = k}^{n} \binom{n}{p}) / 2^n$ quickly? I was solving a question which technically reduces to the following     Given $N$ items, it is equiprobable for each item to be good or bad, what is the probability that number of good items in the set are greater than or equal to $K$.   This can be reduced to $\dfrac{x}{2^n}$ where $\displaystyle x = \sum_{p = k}^{n} \binom{n}{p}$. Is there a more simplified form which is easier to calculate for large values of $N, K$?  Note: It may be safe to assume that we do not require extremely high precision while calculating(first 5-10 digits only).  Thanks! When ever I need to kill a background process I do ps -e | grep &lt;process_name&gt;  Which prints something like this 1766 ? 00:00:13 conky , Then I use the process ID to kill it like so kill 1766 .   Is there any way I can simplify this ? Make it quicker ? reduce the amount of typing ? How does a pharmacy discount card work? Prevent Windows 10 from installing/updating a specific driver? It seems that the recent Synaptics driver update for Synaptics SMBus TouchPad on Windows 10 (build 100049) somehow doesn't see my middle-button. Uninstalling, and installing an older driver for Windows 8.1 from the HP site (my machine is HP EliteBook 8570w) restores the middle button functionality.    Few days later, the driver updates itself again, and I lose the middle-button.  I realize I'm running unsupported software, but is it possible to prevent updating this particular driver, without either disabling updates altogether or preventing other drivers from updating? If string is stretched just by weight, where does the gravitational potential energy goes if only half is converted to elastic potential energy? If a spring is stretched by a weight of mass m, (so the extension is $\Delta x$) then $ k = \frac W{\Delta x} = \frac {mg}{\Delta x}$. So $ k\Delta x= mg $.  When the Spring is stretched by distance $\Delta x$ (by the weight) then it looses gravitational potential enegry. ($ \Delta GPE= mg\Delta x$)  But, when we calculate the change in elastic potential energy, We get  $ U = \frac 12 k(\Delta x)^2 $. Since $ k\Delta x = mg$, $U=\frac 12 mg\Delta x$  At, equilibrium we have no kinetic energy $E_k = 0$  Doesn't that violate Energy Conservation?  Where does the rest of lost GPE goes?  Here's a illustration