Can we ever have limsup$(x_n)= - \infty $?

Question

Let $(x_n)$ be a sequence in $(-\infty, \infty]$.

Could we define the sequence $(x_n)$ so that limsup$(x_n) = -\infty$?

My intuitive thought is no, but I’m not 100% sure.

Answer 1

Consider $(x_{n})$ defined by $x_{n} = -n$. Then $$ \limsup_{n \to \infty} x_{n} = \lim_{n \to \infty} \left( \sup_{m \geq n} \ (-m) \right) = \lim_{n \to \infty} \ ( -n ) = -\infty. $$

Answer 2

Recall that when a sequence $a_n$ has limit $L\in \mathbb{\bar R}$ the following holds

$$\lim a_n=\liminf a_n=\limsup a_n=L$$

therefore for any $a_n \to -\infty$ we have $\limsup a_n=-\infty$.