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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

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  • $\begingroup$ What is the question ? $\endgroup$
    – Kasper
    Mar 1, 2014 at 15:47
  • $\begingroup$ @Kasper It's in the title. Not that it shouldn't be in the post, obviously. $\endgroup$
    – Marcin Łoś
    Mar 1, 2014 at 15:47
  • $\begingroup$ prove that the statement is true using the definition of lim inf $\endgroup$
    – Megan
    Mar 1, 2014 at 15:51
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    $\begingroup$ The contrapositive may be easier to handle. $\endgroup$
    – David Mitra
    Mar 1, 2014 at 15:54
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    $\begingroup$ @Megan People prefer it if you write math using $\LaTeX$ $\endgroup$
    – Kasper
    Mar 1, 2014 at 16:04

1 Answer 1

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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?

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  • $\begingroup$ I'm still a little confused. Thanks anyway. :) $\endgroup$
    – Megan
    Mar 1, 2014 at 23:50
  • $\begingroup$ What are you confused about? Try to formulate your difficulty so that we can help address it. $\endgroup$
    – Joshua Pepper
    Mar 2, 2014 at 0:07

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