Proving that if $\lim\limits_{n\to\infty}a_n=+\infty$ and $\{b_n\}$ is a bounded sequence, then $\lim\limits_{n\to\infty}(a_n+b_n)=+\infty$ I am trying to prove the following problem

If $\lim\limits_{n\to\infty}a_n=+\infty$ and $\{b_n\}$ is a bounded sequence, then  $\lim\limits_{n\to\infty}(a_n+b_n)=+\infty$

I have these definitions as tools;

Definition 1. The sequence $\{a_n\}\to \infty$ if $\forall \;M\in R,\;\;\exists\;n_0=n_0(M)\in N$  $\ni$
                $$n\geq n_{0}\;\implies\;a_n>M.$$
Definition 2. The sequence $\{b_n\}$ is said to be bounded if there exists $M>0,$ $\ni$
                $$|b_n|\leq M \;\forall \;n\geq 1.$$

Can anyone help me out?

Let $B$ be the bound for $(b_n)$. For any $M\gt0$ choose $n_0\in\mathbb N$ such that $n\ge n_0 \implies a_n\gt M+B$.  Then $a_n+b_n\gt M$...
