Continuity & Properties of functions

[ricardianequiva]

Homework Statement

Let f:R->R be a continuous function where limit as x goes to positive/negative infinity is negative infinity. Prove that f has a maximum value on R.

Homework Equations

None

The Attempt at a Solution

I tried to use the definition of infinite limits but I'm not sure how to do this explicitly.

 

[HallsofIvy]

Let X be a real number. Since $\lim_{x\rightarrow \infty}= -\infty$, there exist x_{1[/sup] such that if x> x1, f(x)< X. Since $\lim_{x\rightarrow -\infty}= -\infty$, there exist x2 such that if x< x2, f(x)< X. Now look at the values of f(x) on the closed and bounded interval [x1, x2].}