If a function's limit is zero at infinity what can we say about its supremum?

Question

For example we have a real valued function $f$ defined on a suitable set $A\subset \mathbb{R}$ with $\lim\limits_{x \to \infty }f(x)=0$, then is $\sup\limits_{x\in A}f(x)$ finite or not? Under which conditions this supremum is finite or infinite?

Answer 1

No. Define$$\begin{array}{rccc}f\colon&(0,+\infty)&\longrightarrow&\mathbb R\\&x&\mapsto&\frac1x.\end{array}$$Then $\lim_{x\to+\infty}f(x)=0$ and $f$ has no supremum (in $\mathbb R$).

However, if $f$ is continuous and if the domain is an interval of the type $[a,+\infty)$, then, yes, $f$ has a real supremum. Indeed, since $\lim_{x\to+\infty}f(x)=0$, there is a $M>a$ such that$$x>M\Longrightarrow f(x)<1.$$ On the other hand, since $f$ is continuous and $[a,M]$ is a closed and bounded interval, $f|_{[a,M]}$ is bounded an, in particular, there is a $K>0$ such that $(\forall x\in[a,M]):f(x)<K$. Therefore$$(\forall x\in[a,M]):f(x)<\max\{K,1\}$$and so $\sup f\leqslant\max\{K,1\}$.