Uniform convergence and continuity proof

Question

I was looking at Rudin's proof about uniform convergence and continuity just now, and I wonder how he proceeds from (18) to the next line. Thanks!

Add up to the question just now - in the later part of the proof from equation 19 to 22, how does end up at the final equation without caring the particular $n$ he chooses? (i.e. since this $n$ is a special one, why is he able to make a generalized claim of $\mid f(t) - A\mid \leq \epsilon$ at last?)

Answer 1

$\{f_n\}$ converges to $f$ means $n,m>N \implies |f_n(t) - f_m(t)| < \epsilon/3$

$\lim_\limits{t\to x} f_n(t) = A_n$ means $|t-x| < \delta \implies |f_n(t) - A_n|<\epsilon/3$

$|A_n-A_m| = |A_n - f_n(t) + f_n(t) - A_m + f_m(t) - f_m(t)| \\ |A_n-A_m| <|A_n - f_n(t)| + |A_m - f_m(t)| + | f_n(t) - f_m(t)| < 3\frac{\epsilon}{3} <\epsilon$