If $\{x_n\}\rightarrow \infty$ and $\{y_n\}$ is bounded then $\{x_n+y_n\}\rightarrow \infty$

Question

I'm working on proving the above statement, here's what I have so far.

Proof. Suppose $|x_n+y_n|$ is bounded. Then $\exists \ \varepsilon>0$, such that $|x_n+y_n|<\varepsilon$ for all $n$. But $x_n$ is unbounded, so $\forall \ M>0$, $\exists \ N\in \mathbb{N}$ such that for all $n\geq N$ implies $x_n>M$. Pick $M>K$, so that if $n\geq N$
$$|y_n|=|(y_n+x_n)-x_n|=|x_n-(x_n+y_n)|\geq |x_n|-|x_n+y_n|>M-K$$

I'm looking for a contradiction, and I don't know where to go from here. I have spent too long on this problem as is, so any help is appreciated.

Answer 1

You can construct a simple proof using comparison test. Suppose $|y_n| < M$. Then: $$ x_n + y_n > x_n- M \to \infty $$