Real analysis Limits and continuous functions

Suppose that $f:\mathbb{R}\to \mathbb{R}$ is continuous on $\mathbb{R}$ and that $$
\lim_{x\to -\infty} f(x)=\lim_{x\to +\infty} f(x) =k$$ Prove that $f$ is bounded and if there exist a point $x_0 \in\mathbb{R}$ such that $f(x_0)>k$, then $f$ attains a maximum value on $\mathbb{R}$.

Edit by non OP. The OP seems to be a new user that posted the same question twice in less than 2 hours, both on MSE.
Real analysis continuous functions