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

@Kasper It's in the title. Not that it shouldn't be in the post, obviously. — Marcin Łoś, Mar 01 '14 at 15:47
prove that the statement is true using the definition of lim inf — Megan, Mar 01 '14 at 15:51
Is it similar to the proof of lim sup an as n tends to ∞ =-∞ implies lim an as n tends to ∞ = -∞ ? — Megan, Mar 01 '14 at 16:01

score 0 · Answer 1 · answered Mar 01 '14 at 17:09

0

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?

answered Mar 01 '14 at 17:09

Joshua Pepper

I'm still a little confused. Thanks anyway. :) – Megan Mar 01 '14 at 23:50
What are you confused about? Try to formulate your difficulty so that we can help address it. – Joshua Pepper Mar 02 '14 at 00:07

1 Answers1