limits - $lim_{n rightarrow infty} a_n = +infty, lim_{n rightarrow infty} b_n = +infty$ and $lim_{n rightarrow infty}(a_n + b_n ) = -infty$.

Give an example $\lim_{n \rightarrow \infty} a_n = +\infty, \lim_{n \rightarrow \infty} b_n = +\infty$ and $\lim_{n \rightarrow \infty}(a_n + b_n ) = -\infty$.

I think it's impossible, but my teacher says it's real

Answer

If $\lim_{n \rightarrow \infty} a_n = +\infty$ and $ \lim_{n \rightarrow \infty} b_n = +\infty$, then there is $N \in \mathbb N$ such that

$a_n,b_n> 0$ for $n>N$. Therefore $a_n+b_n> 0$ for $n>N$.

Hence we can not have that $\lim_{n \rightarrow \infty}(a_n + b_n ) = -\infty$.