The Weierstrass approximation theorem says that all continuous functions on $[0,1]$ can be approximated uniformly by polynomials.
Trying to facilitate the digestion of the fatty Christmas food, I was browsing through some complex analysis material, and I found that there is another theorem that states that for any infinite sequence of holomorphic functions that converges uniformly to a function $f$, the resulting function $f$ is holomorphic.
I am having trouble combining the two. All polynomials are entire, so a sequence of polynomials converges to a holomorphic function. However, by Weierstrass, a sequence of real polynomials, which can be seen as restrictions of their complex counterparts, can converge to a non real-differential function, such as $f(x)=|x|$. But I fail to see how a holomorphic function can behave as $|x|$ on the real line.
What am I missing here?