What's wrong with this solution to this Measure Theory problem?

Problem:Let ($X,\mathcal{A},\mu$) be a measure space and suppose $\mu$ is $\sigma$-finite. Suppose $f$ is integrable. Prove that given $\epsilon$ there exist $\delta$ such that

$$\int_A |f(x)|\mu(dx) \;\;<\;\;\epsilon$$

whenever $\mu(A) < \delta$.

Solution: Let $e$ be given, chose $s=\frac{e}{sup(f(x); x\in A)}$, this must be finite because $f$ is integrable.$\int_A |f(x)|\mu(dx)\le \mu(A)*sup(f(x); x\in A)\lt e$

Comments: This is obviously wrong. But I can't find what so wrong about it; is the second inequality invalid, due I need more conditions for it? Thank you!