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**Drexel28** Ah! This is a better question! This is why I asked what norm you were using. The canonical norm on $\displaystyle C[a,b]$ is the infinity norm (BECAUSE it makes it complete), but it is not complete as you mentioned with respect to the $\displaystyle L^p$-norm. So, to answer what I think is your question, which I think is asking does a sequence in $\displaystyle C[a,b]\subseteq L^p[a,b]$ converge to an element of $\displaystyle C[a,b]$ always (since $\displaystyle L^p[a,b]$ is complete we know it converges to an element of $\displaystyle L^p[a,b]$, but will this guy live in $\displaystyle C[a,b])$, the answer is no. We enlarged our space precisely because this isn't true.

Is this what you were looking for?