Pointwise and Uniform Convergence of a Sequence of Real-Valued Functions

In the setting of The Real Numbers: Standing Notation and Background, let $D$ be a set, let $S\subseteq D$, let $(f_{k})_{k\in\mathbb{N}}$ be a sequence in the set of functions from $D$ to $\mathbb{R}$, and let $f:D\to\mathbb{R}$. Write $|t|$ for the absolute value of $t\in\mathbb{R}$.

1. (Pointwise convergence)¶ The sequence $(f_{k})_{k\in\mathbb{N}}$ converges pointwise to $f$ on $S$ if for every $x\in S$ the sequence of real numbers $(f_{k}(x))_{k\in\mathbb{N}}$ converges to $f(x)$.

2. (Uniform convergence)¶ The sequence $(f_{k})_{k\in\mathbb{N}}$ converges uniformly to $f$ on $S$ if for every real $\varepsilon>0$ there is $K\in\mathbb{N}$ such that

for every $k\in\mathbb{N}$ with $K\le k$ and every $x\in S$.