Question 1
Let $f:\mathbb R\to(0,+\infty)$ be continuous and satisfy $\lim_{x\to-\infty}f(x)=\lim_{x\to+\infty}f(x)=0$.
(a) Prove that $f$ attains a greatest value on $\mathbb R$.
(b) Prove that there exist sequences $(x_n),(y_n)$ with $x_n\lt y_n$ for every $n\ge1$, converging to the same limit, such that $f(x_n)=f(y_n)$ for every $n$.
(a) Prove that $f$ attains a greatest value on $\mathbb R$.
(b) Prove that there exist sequences $(x_n),(y_n)$ with $x_n\lt y_n$ for every $n\ge1$, converging to the same limit, such that $f(x_n)=f(y_n)$ for every $n$.