Pointwise convergence, uniform convergence, and the uniformly Cauchy condition for sequences of real-valued functions

Let $X$ be a set and, for each $k\in \mathbb{N}$, let $f_k:X\to \mathbb{R}$ be a real-valued function (The vector space $F^X$ of all functions $X\to F$ with pointwise operations, and $F^n$ as the case $X=n=\{0,1,\dots ,n-1\}$). Let $f:X\to \mathbb{R}$.

The sequence $(f_k)$ converges pointwise to $f$ on $X$ when, for every $x\in X$, the real sequence $(f_k(x))$ converges to $f(x)$ (Limits and Cauchy sequences of reals). Thus the index after which $\mathrm{\mid }{f}_k(x)-f(x)\mathrm{\mid }<\epsilon$ may depend on both $\epsilon$ and $x$.

The sequence $(f_k)$ converges uniformly to $f$ on $X$ when

$(\mathrm{\forall }\epsilon >0)(\mathrm{\exists }N\in \mathbb{N})(\mathrm{\forall }k\ge N)(\mathrm{\forall }x\in X)\mathrm{\mid }{f}_k(x)-f(x)\mathrm{\mid }<\epsilon ,$

where $\epsilon$ ranges over the positive reals. Here one index $N$ serves every point of $X$.

The sequence $(f_k)$ is uniformly Cauchy on $X$ when

$(\mathrm{\forall }\epsilon >0)(\mathrm{\exists }N\in \mathbb{N})(\mathrm{\forall }m,n\ge N)(\mathrm{\forall }x\in X)\mathrm{\mid }{f}_m(x)-f_n(x)\mathrm{\mid }<\epsilon \mathrm{.}$

For each of the three notions above, restricting the error to positive rationals gives an equivalent condition. The real-error condition immediately implies the rational-error condition. Conversely, given a real $\eta >0$, choose $n\ge 1$ with $1\mathrm{/}n<\eta$ by For every $\epsilon >0$ in a complete ordered field there is a natural $n\ge 1$ with $1\mathrm{/}n<\epsilon$; the condition for the positive rational $1\mathrm{/}n$ implies the condition for $\eta$. The real-error form is used because it makes the uniform quantifiers transparent.