Question:
Let $(a_n)_{n\in\mathbb N}$ and $(b_n)_{n\in\mathbb N}$ be sequences with real values. Let $\lim_{n\to\infty}(a_n)=\infty$ and $(b_n)_{n\in\mathbb N}$ be bounded. Prove that $\lim_{n\to\infty}(a_n+b_n)=\infty$.
Proof:
Since we know:
$\lim_{n\to\infty}(a_n)=\infty\space$ $\Leftrightarrow\space$ $\forall K\in\mathbb R^+ \space \space\exists N_1\in\mathbb N $: $n\gt N_1 \space$ $\Rightarrow a_n\gt 2K.$
and $(b_n)_{n\in\mathbb N}$ is bounded $\Leftrightarrow\space$ $\exists K\in\mathbb R^+ \space $ such that $\space \forall n\in\mathbb N \space$ $ |b_n|\le K.$
Now taking $N=N_1$:
$\Rightarrow$ $a_n\gt 2K$ and $|b_n|\le K$
$\Leftrightarrow$ $2K\lt a_n$ and $-K\le b_n \le K$
$\Rightarrow$ $a_n+b_n \gt 2K-K=K$ $\space \forall n\gt N$
Hence $\lim_{n\to\infty}(a_n+b_n)=\infty$
This is an optional question from my Analysis I course, thought it would be a fun little proof. Be great if anyone could check what I have done -struggling to get to grips with these types of proofs.