Suppose I have a sequence of functions, $f_n:\mathbb{R}\rightarrow\mathbb{R}$, each of which is continuous about a value $a^*$. Suppose I define a sequence $\{a_n\}_{n \in \mathbb{N}}$ which tends to $a^*$. Is it true that:
$$\lim_{a\rightarrow a^*}\lim_{n\rightarrow \infty}f_n(a)=\lim_{n\rightarrow \infty}f_n(a_n)$$
if both of the above limits exist. If so why? If not why not?
Not necessarily. Let $f_n(x) = x^n$, let $a=1$, and let $a_n = 2^{1/n}$. Then $f_n(a) = 1$ for all $n$, so the limit on the left is $1$. But $f_n(a_n) = 2$, so the limit on the right is $2$.