Proving that f is bounded on R

[vikkisut88]

Homework Statement

Suppose that f: R -> R is continuous on R and that lim (x -> $$\infty$$+)(f(x) = 0) and lim (x -> $$\infty$$-)(f(x)=0).
Prove that f is bounded on R

Homework Equations

I have got the proof of when f is continuous on [a,b] then f is bounded on[a,b] but I'm unsure as to whether I can use this proof for this particular question.

The Attempt at a Solution

My attempt is merely the proof I speak of from above.

 

[vikkisut88]

but I can't just assume it is that specific function surely? plus i have to prove it's bounded, not unbounded?!?

 

[yyat]

Homework Statement

Suppose that f: R -> R is continuous on R and that lim (x -> $$\infty$$+)(f(x) = 0) and lim (x -> $$\infty$$-)(f(x)=0).
Prove that f is bounded on R

Using the definitions of the above limits you should be able to show that f is bounded on sets of the form [b,+$$\infty$$) and (-$$\infty$$,a] for some a, b.

Homework Equations

I have got the proof of when f is continuous on [a,b] then f is bounded on[a,b] but I'm unsure as to whether I can use this proof for this particular question.

Then use the result for compact intervals [a,b] to complete the proof that f is bounded on all of R.