definition of a limit

**dori1123** · November 29th 2009, 06:24 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, 08:32 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.

Thread: definition of a limit

LinkBack

Thread Tools

Display

definition of a limit

Similar Math Help Forum Discussions

limit definition

use limit definition to prove that the limit is correct

Definition of Limit

use LIMIT definition

1. find the limit 2. use the definition of the limit to prove that

Search Tags