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

Question

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 1

Let $M > 0$ be given, there exits $N_1, N_2 \in \mathbb{N}$ such that:

$a_n > \dfrac{M}{2}$ if $n > N_1$, and $b_n > \dfrac{M}{2}$ if $n > N_2$. Choose $N_0 = \text{max}\left(N_1,N_2\right)$, then if $n > N_0$ then $a_n+b_n > \dfrac{M}{2} +\dfrac{M}{2} = M$, proving $a_n+b_n \to +\infty$ when $n \to +\infty$.

Answer 2

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 $\lim_{n \rightarrow \infty}(a_n + b_n ) = -\infty$.