definition of a limit

**dori1123** · November 29th 2009, 05:24 PM

Suppose $g:[0,\infty)\rightarrow\mathbb{R}$ and $L\in[-\infty,\infty]$ . Prove that $lim_{t\rightarrow\infty}g(t)=L \iff$ for every increasing sequence $(a_n)$ in $[0,\infty)$ , if $a_n\rightarrow\infty$ , then $g(a_n) \rightarrow L$ .

I'm trying to prove [<==] when $L=\infty$ by contradiction. But not sure about the definition:
$lim_{t\rightarrow\infty}g(t)=\infty$ means that $\forall M>0$ , there is $x\in[0,\infty)$ such that $\forall t>x, g(t)>M.$ Is this the correct definition?
So if $lim_{t\rightarrow\infty}g(t)\not=\infty$ , then there exists $M>0$ such that $\forall x\in[0,\infty)$ there exists $t>x$ such that $g(t) \leq M.$ Right?

**roninpro** · November 29th 2009, 07:32 PM

Hello.

Your definition seems to be correct.

Now, you can try for a contradiction/contrapositive by constructing an increasing sequence $\{a_n\}$ which diverges to infinity but $\{g(a_n)\}$ does not.

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