Let $\mathcal R$ and $\mathcal I$ denote the real and imaginary part of a complex-valued function $f: \mathbb{R} \to \mathbb C$.
Is the following claim true?
Let $f_n = (\mathcal{R}_n + i \mathcal{I}_n)$, then $f_n$ converges to $f = \mathcal{R} + i \mathcal{I}$ if and only if $\mathcal{R}_n \to \mathcal{R}$ and $\mathcal{I}_n \to \mathcal{I}$.
I think it is true because it follows the fact that (z_n) Converges IFF (Re(z_n)) and (Im(z_n)) Converges
At each point of the complex-valued function value converges iff the real part and the imaginary part at that particular point converge.
Then, we can use it to conclude that if a complex-valued function is pointwise convergent iff the real part and imaginary part is pointwise convergent.
Is my reasoning correct?
Any thoughts are appreciated!
Yes, this is true. The imaginary part function is continuous, so if $f_n$ converges than so does $im(f_n)$ (and same for the real part).
The other direction requires a bit of $\delta-\epsilon$ but is pretty straightforward to prove.