# Math Help - Continuous functions and boundedness

1. ## Continuous functions and boundedness

Let be f : [a,inf)-->R is is continuous function and suppose that { limf(x) x-->inf } exist.Prove that f is boundedon [a,inf) so f is bounded function.

2. Since the limit exists (say it's $L$) there is an integer $M$ such that if $x>M$ we get $\vert f(x)-L \vert <1$ and using the triangle inequality we get $\vert f(x) \vert < 1+\vert L \vert$. Now consider $f:[a,M] \rightarrow \mathbb{R}$ since it's cont. on a compact set so it attains a maximum and a minimum (say $A,B$ resp ) then $\vert f(x) \vert < \max \{ 1+\vert L \vert ,\vert A\vert ,\vert B \vert \}$ on $[a, \infty )$