Let the positive and continuous $f : \mathbb{R} \to \mathbb{R}$ . Also it is given that
$$ \lim_{x\to\infty} f(x) = 0 $$ $$ \lim_{x\to -\infty} f(x) = 0 $$ I need to prove/show that there is a $y \in \mathbb{R}$ that $f(y) \geq f(x)$ for every $x \in\mathbb{R}$
Assume without loss of generality that $f\neq 0$, and so there is $x_0 \in \mathbb{R}$ such that $f(x_0) > 0$. Let $\epsilon = f(x_0)/2$, then there is a $M > 0$ such that $$ |x| > M \Rightarrow f(x) < \epsilon $$ (This is because $\lim_{x\to \pm\infty} f(x) = 0$) Choose $M$ large enough so that $x_0 \in [-M,M]$. Now, $f$ is continuous on a compact set $[-M,M]$, so there is $y \in [-M,M]$ such that $$ f(y) \geq f(x) \quad\forall x \in [-M,M] $$ And for any $x \notin [-M,M]$, we know that $$ f(x) < \epsilon = f(x_0)/2 < f(x_0) \leq f(y) $$