Proof of theorem describing decay of convolution at infinity?

Question

First question: Is that inequality just cauchy schwarz inequality?

Second question: What happened on that last step, where $g(x-t) \rightarrow g(x)?$

Answer 1

The first inequality is the triangle inequality. That inequality states that the absolute value of a sum is less than or equal to the sum of absolute value. It applies to integrals (which are generalized sums).

The last step is the result of changing the variable $x \rightarrow x-t$ in the inner integral.

Answer 2

The inequality in the proof is just the inequality $|\int h(t)| dt \leq \int |h(t)|dt$ (which has no name). The fact that $\int |g(x-t)| dx=\int |g(x)|dx$ is obtained by making the substitution $z=x-t$. [$\int |g(x)|dx$ is same as $\int |g(z)|dz$]. In measure thoretic language this property is called translation invariance of Lebesgue measure].