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Let $a$ and $b$ be positive constants, and let $u(t)$ be a differentiable function on $[0,\infty)$ satisfying the inequality

$u'(t)\leq au(t)$, $u(0)\leq b$.

Find an upper bound on $u(t)$ and prove that it is the best possible.

My attempt: Using separation of variables, we can see that $u\leq Ce^{at}$ for some constant $C$. I'm not sure how this could possibly bound $u(t)$ given that $a>0$. Am I making a mistake?

Any help appreciated!

Side note: I realize there is a similar question asked, but the answers don't seem to actually answer the question, at least not as far as I can understand.

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    Actually you should arrive at $$u(t)\leqslant be^{at}$$ and the RHS is the desired upper bound. – 2017-02-25

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Use the appropriate integrating factor and integrate $$ (e^{-at}u(t))'=e^{-at}(u'-au)\le 0 $$ from $0$ to $t$.