I'm reading a book on analysis and there's a few steps that I can't understand. It's claimed that if $\mathbf{x},\mathbf{y} \in C[a,b]$, then $|\mathbf{x}(t)-\mathbf{y}(t)|^p$ is Riemann integrable over the interval $[a,b]$. It's not clear to me why. I've tried searching for a proof but to no avail.
Another step is that if $$\int_a^b|\mathbf{x}(t)-\mathbf{y}(t)|^pdt=0$$
where the above is a Riemann integral, then $\mathbf{x}(t)=\mathbf{y}(t)$ for all $t\in[0,1]$. This also isn't clear. Would appreciate any help/explanation regarding this!
It is a known fact that continuous function on a compact interval is Riemmann integrable.
The function $|x(t)-y(t)|^p$ is a continuous function $[a,b]$ as a composition of continuous functions thus Riemman integrable.
Also if the Riemman integral on $[a,b]$ of a non-negative continuous function $h$ is zero then $h=0$ everywhere on $[a,b]$