Question 169
Let $f: \mathbb{R} \to (0; +\infty)$ be a continuous function such that $\lim_{x \to -\infty} f(x) = \lim_{x \to +\infty} f(x) = 0$.
a) Prove that $f(x)$ obtains the maximum value on $\mathbb{R}$.
b) Prove that there exist two sequences $(x_n), (y_n)$ with $x_n \lt y_n$ for all positive integers $n$ such that they have the same limit when $n$ tends to infinity and $f(x_n) = f(y_n)$ for all $n$.
a) Prove that $f(x)$ obtains the maximum value on $\mathbb{R}$.
b) Prove that there exist two sequences $(x_n), (y_n)$ with $x_n \lt y_n$ for all positive integers $n$ such that they have the same limit when $n$ tends to infinity and $f(x_n) = f(y_n)$ for all $n$.