Show, using the definitions, that $\lim_{n \rightarrow \infty} \inf a_n = \infty$ implies that $\lim_{n \rightarrow \infty} a_n = \infty$

Question

Show, using the definitions, that $\lim_{n \rightarrow \infty} \inf a_n = \infty$ implies that $\lim_{n \rightarrow \infty} a_n = \infty$

Here is the definition of $\liminf$ is

$\liminf_{n \rightarrow \infty} a_n = \lim(\inf{a_k : k \geq n})$ where $a_n$ is a sequence

Answer 1

Hint: Observe that: $\displaystyle \inf_{k\geq n} a_k \leq a_n $ (why?)

What does your assumption tell you about the LHS of this inequality?