In Reed and Simon's book on functional analysis they prove that any closed subalgebra $B$ of $C_\mathbb{R}(X)$ with $1 \in B$ is a lattice. Their proof is short and understandable, but there is one inequality that I am having trouble understanding.
Let $f \in B$ and WLOG assume $\|f\|_\infty \leq 1$. By the Weierstrass approximation theorem there exists a sequence of polynomials $P_n$ that converges uniformly to $|x|$ on $[-1, 1]$, i.e. $$\big| P_n(x) - |x| \big| \leq 1/n \quad \forall x \in [-1,1]. \tag{1}$$ What I have not understood is how they use the above to say that since $\|f\|_\infty \leq 1$, it follows that $$\|P_n(f) - |f| \|_\infty < 1/n. \tag{2}$$ How do we go from (1) to (2)?
My thoughts are that by definition of the $\|\cdot\|_\infty$ norm, we have that $$\|P_n(f) - |f| \|_\infty = \sup_{x \in X} \big|P_n(f(x)) - |f(x)|\big|.$$ But I am having trouble relating this to (1).