Question

Here is my question.

I have two continuous functions $f_n(x)$ and $a_n(x)$. I know for a fact that $\lim_{n\rightarrow\infty}a_n(x) = a(x)$, and I would like to perform a partial application of the limit to the composition of functions, i.e. something like

$$ \lim_{n\rightarrow\infty}f_n(a_n(x)) =(?) \lim_{n\rightarrow\infty} f_n(a(x)) $$

My gut feeling is that in order for this to hold, I need $f_n(x)$ to satisfy uniform convergence, which I can obtain for $f_n$ through an application of Dini's theorem as the function meets the requirements.

The reason I want to take the limit only partially is because I want to rely on regular variation properties of $f_n$ later on.

It would be great if any of you could confirm/infirm.

Thanks and cheers!

Answer 1

If

• $g_n \colon A \to B$ and $g \colon A \to B$,

• $f_n \colon B \to C$ are continuous,

• $f_n \to f$ uniformly on $B$, and

• $g_n \to g$ pointwise on $A$,

then for each fixed $x \in A$ we have

$$ \lim_{n \to \infty} f_n(g_n(x)) = f(g(x)) $$

since $g_n(x)$ is a convergent sequence in $B$ (see exercise 9 of chapter 7 in Rudin's Principles of Mathematical Analysis). The fact that

$$ \lim_{n\to\infty} f_n(g(x)) = f(g(x)) $$

follows from the pointwise convergence of $f_n \to f$. Your result follows.