Let $(f_ {n})$ be a sequence of functions from $\mathbb{R}$ to $\mathbb{R}$. Prove that
$\displaystyle\{x\in X:(f_{n}(x))\text{ converges in}\ \mathbb{R}\}=\bigcap_{k=1}^{\infty} \bigcup_{n=1}^{\infty} \bigcap_{p=1}^{\infty} \{x\in X:|f_{n}(x)-f_{n+p}(x)|<\frac{1}{k}\}$
My attempt was
$(f_{n}(x))\text{ converges}$
iff $\forall \epsilon>0:\exists n\in \mathbb{N}:\forall m\in \mathbb{N}: m>n\rightarrow |f_n(x)-f_m(x)|<\epsilon$
iff $\forall k\in \mathbb{N}:\exists n\in \mathbb{N}:\forall p\in \mathbb{N}:|f_n(x)-f_{n+p}(x)|<\frac{1}{k}$
iff $\displaystyle x\in\bigcap_{k=1}^{\infty} \bigcup_{n=1}^{\infty} \bigcap_{p=1}^{\infty} \{x\in X:|f_{n}(x)-f_{n+p}(x)|<\frac{1}{k}\}$
I did this, but my teacher told me that I was not demonstrating anything with this. It's wrong? I only used the Cauchy criteria and the archimedean property.