Reference request: proof of completeness of the space $L_p$

Question

Does someone know about a proof for $L_p$ completeness that is not difficult to follow?

I started to watch a YouTube video about it https://www.youtube.com/watch?v=BWnV8fBlDQQ start [10:20] minute. Though I found it unclear to follow the explanation there.

Suggestions are very appreciated.

Answer 1

The lecture notes by John K. Hunter:

https://www.math.ucdavis.edu/~hunter/measure_theory/measure_notes_ch7.pdf