# Proof $a + \infty = \infty$

Given a convergent sequence $(a_n)$ with limit $a \in \mathbb{R}$ and a divergent sequence $(b_n)$ tending to infinity.

I want to prove now using

1. the boundedness of $(a_n)$: $\exists C \in \mathbb{R} \forall n \in \mathbb{N}: |a_n| \leq C$
2. $\forall K > 0 ~\exists N ~\forall n \geq N : b_n > K$

that the sequence $(a_n + b_n)$ is also tending to infinity. This is obviously true, however I can't seem to come up with a consistent solution.

If anyone liked to share a promising approach, I'd be grateful.

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HINT: Given $K>0$, there is an $N\in\Bbb N$ such that $b_n>K+C$ whenever $n\ge N$. For all $n$ we have $a_n+b_n\ge-C+b_n$, so what happens if $b_n>K+C$?