I'm having trouble proving the fact that in first countable topological spaces, a net converges to a point iff every subsequence of the net converges to the same point.
I first encountered this problem when I was learning dominated convergence theorem. I often tried to pass the limit under the integral sign, especially when dealing with partial derivative of an integral and using Leibniz's rule. The statement of DCT involves a sequence of functions and the convergence of a sequence of integrals. $$\lim_{n \rightarrow \infty} \int f_n = \int \lim_{n \rightarrow \infty} f_n$$But when I was dealing with partial derivative of an integral, it's a net of functions which gives the convergence of a net of integrals. For example, $$\lim_{h\rightarrow0} \int_0^t \frac{H(t+h,s)-H(t,s)}{h}ds = \int_0^t \lim_{h\rightarrow 0} \frac{H(t+h,s)-H(t,s)}{h}ds = \int_0^t \frac{\partial H(t,s)}{\partial t} ds$$
This is often proved by DCT.
I don't know why exactly we can do this. I was told it is because of the result that a net converges to a point iff every subsequence of the net converges to the same point in first countable topological spaces. I tried to prove this result by myself but couldn't get anywhere.
Thanks in advance for any help!