Show that if $X_n \to X$ in $L^2$, $X_n$ is cauchy convergent in $L^2$.

Question

This can be shown by using the Minkowski inequality: $$ (E[|X_n - X_m|^2])^{1/2} \leqslant (E[(X_n - X)^2])^{1/2} + (E[(X_m - X)^2])^{1/2} $$ Where both goes to zero by assumption and we have the conclusion.

Two things are not clear to me:

1. What is the difference between the minkowski inequality and the triangle inequality? It seems to me that they're applied similarly really often.
2. What are the mathematical passage used to prove what I wrote?

Answer 1

In your second question, you can prove it formally by "adding zero" as in $$ \|X_n-X_m\| = \|X_n - X + X-X_m\| \le \|X_n-X\| + \|X-X_m\|, $$ where $\|\cdot\|$ denotes $(E[|\cdot|^2])^{1/2}$ the $L^2$-norm. The inequality is an invocation of Minkowski's inequality (equivalently the triangle inequality in the space $L^2$).