Continuous function #1

Quote:

Originally Posted by Also sprach Zarathustra

Prove:

If $f(x)$ is continius function on $[0,\infty)$ and if $lim_{x\to \infty} f(x)=M<\infty$ , then $f(x)$ is bounded on $[0,\infty]$

What I did..

I looked the next two intervals:

1. $[0,M]$ , $f(x)$ is bounded there.

2. $[M, \infty)$ , and I don't know what to do next here...

Thanks!

Use the definition of $\lim_{x\to \infty} f(x)=M$ , taking $\varepsilon=1$ . It tells you that there exists N such that $|f(x)-M|<1$ whenever x>N. So f is bounded on $[N,\infty)$ . But it is also bounded on [0,N] ... .