Proving that function is bounded when its continous and its limits at infinity are bounded.

Question

Let f be a function from real numbers to real numbers and let f(x) be a continous function, such that:

$$\lim_{x\rightarrow\infty}f(x)$$ $$\lim_{x\rightarrow-\infty}f(x)$$

are bounded.

How do I prove that f(x) is bounded?

Answer 1

Let $L_+$ and $L_-$ be the two limits at $\infty$ and $-\infty$.

There exist $M_+$ and $M_-$ such that $|f(x)-L_+| < 1$ for all $x > M_+$, and $|f(x) - L_-|<1$ for all $x < M_-$. Then all you need to do is bound $f$ on the compact interval $[M_-,M_+]$.