Prove or disprove : if $a_n$ has a limit and $b_n$ doesn't have a limit then $a_n + b_n$ doesn't have a limit

Question

I think it's wrong but I couldn't find an example that disproves this. If this is true I need to prove it and if it's wrong I have to give an example to disprove it.

Answer 1

Assume it's true, then

$$\lim_{n\to\infty}b_n=\lim_{n\to\infty}\left(a_n+b_n\right)-\lim_{n\to\infty}a_n$$

and then...

Answer 2

Take $a_n=n$ and $b_n=(-1)^n$. You can choose for $b_n$ any bounded sequence that doesn't converge, and $a_n$ as a diverging sequence.