I want to make sure my understanding of the proof is correct.
For a Cauchy sequence $\{f_n\}$ in $L^p$, we want to find a $f\in L^p$ such that $f_n\stackrel{L^p}\to f$
Now, skipping the technicalities of the proof, if we manage find some $f$, for which $$\forall \epsilon>0\;\exists n_0,\; m\geq n_0 \;\;\|f-f_m\|_{L^p}\leq\epsilon$$ Are we done? I'd say yes, as, $f=\underbrace{(f-f_m)}_{\in L^p}+f_m \in L^p$ and $$\epsilon \geq \|f-f_m\|\geq \Big|\|f\|-\|f_m\| \Big|$$ Implies $\|f_m\|\stackrel{L^p}\to \|f\|$
Am I correct in my reasoning?
Thank you.