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**ragnar** I am asked to prove that, if $\displaystyle $\displaystyle \lim_{n \rightarrow \infty} ||f_{n} - f|| = 0$$ and $\displaystyle $\displaystyle \lim_{n \rightarrow \infty} ||g_{n} - g|| = 0$$, prove that $\displaystyle $\displaystyle \lim_{n \rightarrow \infty} \int_{I} f_{n} \cdot g_{n} = \int_{I}f \cdot g$.$

I doubt that I want to do this by brute force by providing some increasing sequence of upper functions that approach $\displaystyle f \cdot g$, though maybe I do since I know that $\displaystyle \displaystyle \lim_{n \rightarrow \infty} ||f_{n}|| = ||f||$ and likewise for $\displaystyle g$.

In any case, the alternative seems to be to use to the Dominated Convergence Theorem, in which case I want to show that $\displaystyle \{ f_{n} \cdot g_{n} \}$ converges almost everywhere (presumably to $\displaystyle f \cdot g$), and that there is a non-negative function which is integrable and dominates $\displaystyle |f_{n} \cdot g_{n}|$. But the first part seems like it might be false, since I don't know that the interval over which these functions are integrated is bounded.

Maybe I want to try an epsilon-delta argument?