I was reading the following theorem and its proof:
If $(f_n)$ and $(g_n)$ are uniformly convergent sequences of functions, then $(f_n+g_n)$ is a uniformly convergent sequence of functions.
Proof. Let $\epsilon>0$. Then there exists $K\in\mathbb{N}$ such that $k\geq K$ implies $\left \| f_k-f \right \|_A<\epsilon/2$ since $(f_n)$ is uniformly convergent to $f$ on $A$. Similarly, there exists $M\in\mathbb{N}$ such that $m\geq M$ implies $\left \| g_m-g \right \|_A<\epsilon/2$. Set $N=\max\{K,M\}$. It follows that for $n\geq N$, we have $\left \| (f_n+g_n)-(f+g) \right \|_A=\left \| (f_n-f)+(g_n-g) \right \|_A\leq\left \| f_n-f \right \|_A+\left \| g_n-g \right \|_A<\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon.$ Therefore, $(f_n+g_n)$ is uniformly convergent to $(f+g)$ on $A$. $\square$
My quesiton is: What is the meaning of those double absolute value bars, and what do their subscripts denote? Thanks!