Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with $f(0)>0$ and $$\lim_{x \to \infty} f(x)= \lim_{x \to -\infty} f(x)=0$$
$(i)$ Show that $f$ is bounded.
$(ii)$ Let $A=sup\{f(x):x \in \mathbb{R} \}$ show that there is a point $x_0 \in \mathbb{R}$ such that $f(x_0)=A$
My Attempt
$(i)$ I am not really sure how to approach this as the question seems intuitive if you draw a sketch or think about it. But proving it rigorously is what I am struggling with. Start with definitions;
$\bullet$ $\lim_{x \to \infty} f(x)=0$ means $$\forall \ \ \epsilon>0 \ \ \exists \ \ N \ \ s.t \ \ |f(x)|<\epsilon \ \ for \ \ x>N$$
$\bullet$ $\lim_{x \to -\infty} f(x)=0$ means $$\forall \ \ \epsilon>0 \ \ \exists \ \ N \ \ s.t \ \ |f(x)|<\epsilon \ \ for \ \ x<N$$
but how do I show it is bounded, can't use the EVT as it is not on a closed interval...
any help on $(i) and (ii)$ would be very much appreciated