The following is an excerpt from Rudin's book in mathematical analysis. Here he states:
The part highlighted in red is the one I can't seem to wrap my head around. I thought that if we wanted to know whether the limit, say $f$, of a sequence of functions, say $f_n$, is continuous or not then we would just need: $$\lim_{t\to x} (\lim_{n \to \infty}f_n(t)) = f(x)$$
I.e. just that the limit of functions $f_n$, assumed to be $f$, is continuous by definition. So I don't understand the right hand side of the equation in red. Can somebody explain this?
Best Answer
To ask, whether $f$, given by $f(t) := \lim_n f_n(t)$ for each $t$, is continuous at $x$, is asking whether $$ \tag 1 \lim_{t\to x} f(t) = f(x) $$ Now, let's plug in the definition of $f$ as the limit of the $f_n$ on both sides of (1). We get (as both $f(x) = \lim_n f_n(x)$ and $f(t) = \lim_n f_n(t)$ hold), that $$ \tag 2 \lim_{t\to x} \lim_n f_n(t) = \lim_n f_n(x) $$ As the $f_n$ are continuous by assumption, we may write $f_n(x) = \lim_{t\to x} f_n(t)$ in (2), giving us $$ \tag 3 \lim_{t\to x} \lim_n f_n(t) = \lim_n \lim_{t\to x} f_n(t) $$ which is Rudin's red formula.