Understanding how to justify inequality between limits The problem goes as follows:
Let $f_n$ be a sequence of functions with domain $[0,1]$ and let $B$ be a dense subset of $[0,1]$. Show that if $f_n$ is increasing, for every $n\in\mathbb{N}$ and $(f_n)\rightarrow f$ pointwise in $B$, then $(f_n)\rightarrow f$ in $A=\{x\in[0,1]:f$ is continous in $x\}$
Although I know how to procede on this problem (by showing that $\limsup f_n(x)\leq f(x) \leq \liminf f_n(x)$), I don't know how to procede in proving those inequalities. 
I know that, because $B$ is dense, there are sequences of elements in $B$ $(x_m)$ such that $x_m\rightarrow x^-$ and $(y_m)$ such that $y_m\rightarrow x^+$ for any $x\in A$. From there, since $f_n$ is increasing, 
$$f(x_m)\leq f(x) \leq f(y_m)$$
$$\implies \lim_{n\rightarrow \infty} f_n(x_m)\leq f(x) \leq \lim_{n\rightarrow \infty} f_n(y_m)$$
From here, what is the logic behind $\limsup f_n(x) \leq \lim_{n\rightarrow \infty} f_n(x_m)$ (analogous on the other side of the inequality) to get what I need? My guess would be that
$$\limsup f_n(x) = \lim_{n\rightarrow \infty} f_n(x) = \lim_{m\rightarrow \infty}\lim_{n\rightarrow \infty} f_n(x_m)\leq \lim_{n\rightarrow \infty} f_n(x_m)$$
But I'm really not that sure about it.
 A: It seems the following.

Although I know how to procede on this problem (by showing that $\limsup f_n(x)\leq f(x) \leq \liminf f_n(x)$), I don't know how to procede in proving those inequalities. 

Let $x\in A$ be an arbitrary point of continuity of the function $f$. Because the set $B$ is dense in the segment $[0,1]$, there are a non-decreasing sequence $\{x_m\}$ and a non-increasing sequence $\{y_m\}$ of elements in $B$
which both converge to the point $x$. Since all $x_m, y_m\in B$ and $(f_n)\rightarrow f$ pointwise in $B$, for each index $m$ we have 
$$\lim_{n\rightarrow \infty} f_n(x_m)=f(x_m)$$ and $$\lim_{n\rightarrow \infty} f_n(y_m)=f(y_m).$$ 
Let $n,m$ be arbitrary indices. Since the function $f_n$ is increasing, $f_n(x)\le f_n(y_m)$. Then 
$$\limsup f_n(x)\le \limsup f_n(y_m)= \lim f_n(y_m)=f(y_m).$$ 
Similarly, 
$$\liminf f_n(x)\ge \liminf f_n(x_m)= \lim f_n(x_m)=f(x_m).$$ 
Since $x$ is a point of continuity of the function $f$, we have 
$$\lim_{m\rightarrow \infty} f(x_m)=f(x)=\lim_{m\rightarrow \infty} f(y_m).$$
Thus  
$$\limsup f_n(x)\le \lim_{m\rightarrow \infty} f(y_m)=f(x)$$ and  
$$\liminf f_n(x)\ge \lim_{m\rightarrow \infty} f(x_m)=f(x).$$
