Question

Prove that a sequence that diverges to +∞ (resp. −∞) is divergent

Answer 1

A sequence $\{a_n\}_{n=1}^{+\infty}$ that diverges to +∞ or -∞ is unbounded:

$a_n\to+\infty$ means $\forall M>0\,\exist N\in\mathbb{N}\, \forall n>N a_n>M$ that implies $\sup\limits_n| a_n|=+\infty$

$a_n\to-\infty$ means $\forall M>0\,\exist N\in\mathbb{N}\, \forall n>N a_n<-M$ that implies $\sup\limits_n| a_n|=+\infty$

A sequence $\{a_n\}_{n=1}^{+\infty}$ that converges to a finite number A is bounded:

$\forall \varepsilon>0\,\exist N(\varepsilon)\in\mathbb{N}\, \forall n>N(\varepsilon) |a_n-A|<\varepsilon$

If we take $\varepsilon=1$ then $\forall n>N(1) |a_n-A|<1$, $|a_n|<|A|+1$ and $\sup\limits_n |a_n|\leq \max\{|A|+1,|a_1|, |a_2|,\dots,|a_{N(1)}|\}<+\infty$

Therefore, any sequence that diverges to +∞ or -∞ is divergent.