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