# Math Help - definition of a limit

1. ## definition of a limit

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?

2. 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.