Cauchy sequence in $L^p(\mu)$

Question

I am following Rudin's Real and Complex Analysis. While proving that $L^p(\mu)$ is a complete metric space a theorem is used-

If $1\leq p \leq \infty$ and if $ \{f_n\}$ is a Cauchy sequence in $L^p(\mu)$, with limit $f$ , then $\{f_n\}$ has a subsequence which converges pointwise almost everywhere to $f(x)$.

I understood the proof of the theorem. Now I am trying to find out this but can't proceed -

Is there a sequence such that the whole sequence does not converge pointwise? (i.e. only a proper subsequence converges pointwise but not the sequence itself).

Answer 1

Consider $f_1 = \chi_{[0,1]}$, $f_2 = \chi_{[0,0.5]}$, $f_3 = \chi_{[0.5,1]}$, $f_4 = \chi_{[0,0.25]}$, $f_5 = \chi_{[0.25,0.5]}$, $f_6 = \chi_{[0.5,0.75]}$, $f_7 = \chi_{[0.75,1]}$....

$f_n$ converges to $0$ in $L^1$ but does not converge pointwise anywhere in $[0,1]$.