If a function is complex differentiable, how do we know its real and imaginary parts are infinitely differentiable?

Sorry I'm really rusty on multivariable calc.

Suppose $f: \mathbb{C} \to \mathbb{C}$ is holomorphic and $f(x,y) = u(x,y) + v(x,y)i$, then we know that the partials $u_x, u_y, v_x, v_y$ exist and are continuous, so $u$ and $v$ are real differentiable. But how do we use the infinite complex differentiability of $f$ to prove the infinite real differentiability of $u$ and $v$?

There you have a long way to go, one semester worth of function theory.

First you prove Goursat's theorem.

With that the Cauchy integral theorem.

Then the Cauchy integral formula.

Then that you can develop every holomorphic as a power series.

And every power series is infinitely differentiable inside its domain of convergence.

The power series expansion of $$f$$ in any point $$z=x+iy$$ immediately gives power series expansions of its real and imaginary components in $$(x,y)$$, having a converging power series expansion implies infinite differentiability.

• That's not an answer to OP's question. Aug 5 '20 at 9:48
• @Haldot : How do you figure? That is the program of the first part of complex calculus, the last statement is the claim in question. Aug 5 '20 at 10:04
• OP mentioned infinite differentiability of $f$ as something he already knows. The question was how to deduce from this the fact that $Re f$ and $Im f$ are infinitely differentiable as functions of 2 variables. Aug 5 '20 at 10:07
• Your edit made everything clear, thanks Aug 6 '20 at 8:17