Let {$x_n$} and {$y_n$} be bounded sequences. I want to show that {$x_n + y_n$} is a bounded sequence.

I tried to use definition of limits, but I don't think I can say that {$x_n$} and {$y_n$} converges to x, and y because even if sequences are bounded, they can still be divergent.

What should I do?

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    $\begingroup$ $|x_n|\leq M$ and $|y_n|\leq N$ for some $M,N>0$ and every $n\in\Bbb N$. What about $|x_n+y_n|\leq ???$ $\endgroup$ – Dave Mar 10 '17 at 4:02

Triangle inequality.

Suppose $$ |x_n|<M_1\\ |y_n|<M_2 $$ for any $n$.

Then $$ |x_n|+|y_n|<M_1+M_2\implies |x_n+y_n|\leq |x_n|+|y_n|<M_1+M_2 $$ for any $n$ and we conclude.

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    $\begingroup$ Omg.... It can be like this easy. Thank you. $\endgroup$ – user421044 Mar 10 '17 at 4:08
  • $\begingroup$ @user421044 sure thing. When in doubt, use the triangle inequality $\endgroup$ – qbert Mar 10 '17 at 4:10

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