$\lim_{n \rightarrow \infty} a_n = +\infty, \lim_{n \rightarrow \infty} b_n = +\infty$ 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

• Your teacher is mistaken. Or perhaps one of the limits of $a_{n}, b_{n}$ is $-\infty$. – Paramanand Singh Oct 5 '17 at 7:05

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 $\lim_{n \rightarrow \infty}(a_n + b_n ) = -\infty$.
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 $n \to +\infty$.
• But $\lim_{n \rightarrow \infty} (a_n + b_n) = -\infty$ by condition – Gorogorov Oct 5 '17 at 3:56