Let $I \subset \mathbb{R}$ be an unbounded interval. Assume $\forall a \in I\exists M(a)>0,\, M(a)<\infty$. Can I deduce that $\sup\limits_{a\in I}M(a)<\infty$ as well? Is there any counterexample for unbounded interval case? I know that for bounded interval, I can use compactness argument but I am not sure about unbounded interval case. Here $M$ is just a constant which depends on $a$.
Any help is much appreciated!
Take $I=(0, \infty)$ and $M(a)=a$ for a counterexample. The claim is false even for bounded intervals: take $I=(0,1)$ and $M(a)=\frac 1 a$.