In Rudin there are two relevant theorems. Here is one of them:
6.20 Theorem Let $f$ be Riemann integrable on $[a,b]$. For $a \le x \le b$, put $$F(x)=\int_a^x f(t) \, dt.$$ Then $F$ is continuous on $[a,b]$; furthermore, if $f$ is continuous at a point $x_0$ of $[a,b]$, then $F$ is differentiable at $x_0$, and $$F'(x_0)=f(x_0).$$
You have already stated the other theorem (6.21) in your edit, but you do not need that theorem to answer your question. (In fact, it does not apply here since $f$ may not be continuous.) If $f$ is Riemann integrable then $F$ (as defined in Theorem 6.20) is continuous on $[a,b]$. Then $\lim_{x \rightarrow b} F(x) = F(b)$, i.e. $$\lim_{x \rightarrow b} \int_a^x f(t) \, dt = \int_a^b f(t) \, dt.$$
Your proof states "Let $F$ denote the antiderivative of $f$ which exists by assumption", which is simply not true!