If $\limsup_{n\to\infty} \ x_{n} =-\infty$ then why $\lim_{n\to\infty} \ x_{n} =-\infty$?

Question

I cannot understand why, if $\limsup_{n\to\infty} \ x_{n} =-\infty$ then $\lim_{n\to\infty} \ x_{n} =-\infty$? Can anybody explain it? What's the relationship between $\limsup$ and $\lim$?

Answer 1

The following relation holds: $$\liminf_{n \to \infty} x_n \leq \limsup_{n \to \infty} x_n$$ If $\liminf$ and $\limsup$ exists (in $\mathbb{R}\cup \{-\infty,\infty\})$ and are equal, we define $$ \lim_{x \to \infty} x_n = \liminf_{n \to \infty} x_n = \limsup_{n \to \infty} x_n$$ Therefore, if $\limsup_{n \to \infty} x_n = - \infty$ we must have $\lim_{n \to \infty} x_n = -\infty$.

Answer 2

By definition, $$ \limsup_{n\to\infty} x_n=\lim_{n\to\infty}\sup\{x_n,x_{n+1},\ldots\}, $$ and $$ x_n\leq\sup\{x_n,x_{n+1},\ldots\}. $$