Let {X_n} be a sequence of real numbers such that $lim_{n\rightarrow \infty} X_n =0$. Show that for any bounded sequence{Y_n}, $lim_{n\rightarrow\infty}X_nY_n =0$
Please, help.

Question

Let {X_n} be a sequence of real numbers such that $lim_{n\rightarrow \infty} X_n =0$. Show that for any bounded sequence{Y_n}, $lim_{n\rightarrow\infty}X_nY_n =0$
Please, help.

loser66 · Answer

I have to use Squeeze Theorem to show while I can prove it simpler.
By Limit operation theorem, 
$lim_{n--> \infty}X_nY_n = lim X_n*lim Y_n= 0* y=0$ 
But I don't know how to apply Squeeze

anonymous · Answer

You can't use that limit operation though. You dont know if the $\lim_{n\rightarrow \infty}y_n$ exists or not.

anonymous · Answer

Since the sequence $y_n$ is bounded, we know there exists an $M>0$ such that : $$0\le\left|y_n\right|\le M$$for all $n\in \mathbb{N}$.  Therefore: $$0\le \left|x_ny_n\right|=\left|x_n\right|\left|y_n\right|\le \left|x_n\right|\cdot M.$$Try using the squeeze theorem here.

mathmath333 · Answer

as it is a bounded sequence

u can use  ,$\large\tt \color{black}{Y_n=m_n\times X_n}$ where m is some constant

loser66 · Answer

I am sorry for being late; my computer is stupid

anonymous · Answer

No worries, I'm running around cleaning. Company coming over later.

loser66 · Answer

Thank you so much.