# Let {$x_n$} and {$y_n$} be bounded sequences

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?

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

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.