Assuming the theorem that a continuous real-valued function on a closed

bounded interval is bounded and attains its bounds, prove that if f : R → R is

continuous and f(x) → +∞ as x → ±∞ then there exists some x_0 ∈ R such that

f(x) > f(x_0) for all x ∈ R.

I'm not even sure where to start with this - doesn't anybody have any hints.

Thanks.