Suppose that f : ℝ→ℝ is continuous on ℝ and that lim f =0 as x→ -∞ and lim f =0 as x→∞. Prove that f is bounded on ℝ and attains either a maximum or minimum on ℝ. Give an example to show both a maximum and a minimum need not be attained.
The Attempt at a Solution
The example i plan on using is f(x) = 0. But i am not sure which theorem i am supposed to use. I am thinking the maximum-minimum theorem but i am unsure how to start this without a closed bounded interval. Can anyone help me start?