Q:

Approximating a function in every pair of points in X; clarification of Lemma 4.3.2 of Pedersen's Analysis Now

In Pedersen's Analysis Now book, lemma 4.3.2 says: Let $\mathcal{A}$ be a vector space of continuous, real-valued functions on a compact Hausdorff space $X$. If $f \vee g$ and $f \wedge g$ belong to $\mathcal{A}$ for all $f$ and $g$ in $\mathcal{A}$, then every continuous function on $X$ that can be approximated from $\mathcal{A}$ in every pair of points in $X$ can in fact be approximated uniformly from $\mathcal{A}$. 
My question is what does "approximated from $\mathcal{A}$ in every pair of points in $X$" mean?

A:

It means that, for given $f\in C(X)$ 
$\forall\, x, y\in X$ there are functions $f_n\in \cal{A}$ such that $f_n(x)\rightarrow f(x)$ and $f_n(y)\rightarrow f(y)$

