Consider the metric space $$B = \{ f \in C[0,1] : \int_a^b \left| f(x) \right| dx \leq 1\},$$ where $d(f,g) = \int_0^1 \left| f(x) - g(x) \right|dx$.
I'm trying to show that this metric space is not complete. I have proved that the metric space is not totally bounded. I did this by showing that there existed at least one sequence with a subsequence that was not Cauchy.
I'm aware that a metric space is compact $\Leftrightarrow$ Complete + Totally bounded.
Similarly, I know that compactness $\implies$ completeness, but $\neg$(compact) does not imply $\neg$(complete).
Can't seem to get the logic out.