Interchange of limits and uniform limits Analysis Vol II by Terence Tao
I have no trouble proving the uniform limit theorem using the given hint
$d_Y(f(x),f(x_0)) \le d_Y(f(x),f^{(n)}(x)) + d_Y(f^{(n)}(x),f^{(n)}(x_0)) + d_Y(f^{(n)}(x_0),f(x_0))$
However, this exercise has demanded quite some time and a sleepless night, and i'm afraid the solution is blatantly simple. Anyway:

Propostion 14.3.3 (Intercchange of limits and uniform limits). Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, with Y complete, and let E be a subset of X. Let $(f^{(n)})^\infty_{n=1}$ be a sequence of functions from E to Y, and suppose that this sequence converges uniformly in E to some function $f:E \to Y$. Let $x_0 \in X$ be an adherent point of E, and suppose that for each n the limit $\lim_{x\to x_0; x \in E}f^{(n)}(x)$ exists. Then the limit $\lim_{x\to x_0; x \in E}f(x)$ also exists, and is equal to the limit of the sequence $(\lim_{x\to x_0; x \in E}f^{(n)}(x))^\infty_{n=1}$; in other words we have the interchange of limits
$\lim_{n\to \infty}\lim_{x\to x_0; x\in E} f^{(n)}(x) = \lim_{x\to x_0; x\in E} \lim_{n\to \infty} f^{(n)}(x)$

 A: I went to my professor. 
$\{f^n\}$ converges uniformly to $f$ and every convergent sequence is a Cauchy sequence, thus:
$\forall\epsilon\gt0 \;  \exists N\;\forall x \in E\;\; \forall j,k\ge N, d_Y(f^j(x),f^k(x)) \lt \epsilon /3$.  
We can define $\forall n \lim_{x\to x_0;x\in E} f^n(x) := L_n$, and with this:
$\forall \epsilon \gt 0 \;\forall j, k \ge N \; \exists \delta := \min(\delta_j,\delta_k) \gt 0$ such that  $\forall x\in E (d_X(x,x_0) \lt \delta \implies$both $ d_Y(f^j(x),L_j),\; d_Y(f^k(x),L_k)\lt \epsilon /3$
$\forall \epsilon \;\exists N \;\forall j,k\ge N \quad d_Y(L_j,L_k)\le d_Y(L_j,f^j(x))+ d_Y(f^j(x),f^k(x))+d_Y(f^k(x),L_k) \lt \epsilon $ due to triangle inequality.
And this makes $\{L_n\}$ a Cauchy sequence in $Y$. We are then qualified to define $\lim_{n\to\infty}\lim_{x\to x_0;x\in E}f^n(x)=\lim_{n\to\infty}L_n := L\quad$ And now I claim that
$\lim_{x\to x_0;x\in E}\lim_{n\to\infty}f^n(x) = \lim_{x\to x_0;x\in E}f(x) = L$.
$\forall \epsilon \;\exists N :=\max(M,P)* \implies \exists \delta_N$ such that $\forall x \in E \;(d_X(x,x_0)\lt \delta_N \implies d_Y(f(x),L)\le d_Y(f(x),f^N(x))+d_Y(f^N(x),L_N)+d_Y(L_N,L)\lt \epsilon$
*the M such that $\forall m\ge M\; d_Y(f^m(x),f(x))\lt \epsilon/3$ and P such that $\forall p\ge P \; d_Y(L^p,L)\lt \epsilon /3$
