FALSE: a pointwise limit of a sequence of Riemann integrable functions on $[a,b]$ is Riemann integrable

Given: The set $E:=\mathbb{Q}\cap [0,1]$, a surjection $s:\mathbb{N}\to E$, the finite sets $F_n:=\{s(k):k<n\}$ for $n\in \mathbb{N}$, and the indicators $f_n:[0,1]\to \mathbb{R}$ with $f_n(x)=1$ for $x\in F_n$ and $f_n(x)=0$ otherwise.

The false claim: a pointwise limit of Riemann integrable functions on a closed bounded interval with distinct endpoints is Riemann integrable.

$\mathbb{Q}$ is countably infinite and every subset of an at most countable set is at most countable, so $E$ is at most countable; $E$ is nonempty, since $0\in E$; and a nonempty at most countable set admits a surjection from $\mathbb{N}$ ($\mathbb{Q}$ is countably infinite, Every subset of an at most countable set is at most countable, Finite, countably infinite, countable, uncountable, A nonempty set is at most countable iff it is a surjective image of $\mathbb{N}$).

A bounded function on $[a,b]$ that is continuous at every point other than $r$ listed points is Riemann integrable (A bounded function on $[a,b]$ that is continuous except at finitely many points is Riemann integrable, Lower bound, bounded below, bounded set).

The Dirichlet function restricted to $[0,1]$, that is $g:[0,1]\to \mathbb{R}$ with $g(x)=1$ for rational $x$ and $g(x)=0$ for irrational $x$, is bounded and not Riemann integrable: its lower Darboux integral is $0$ and its upper Darboux integral is $1$ (FALSE: every bounded function on $[a,b]$ is Riemann integrable, The Dirichlet function $1_{\mathbb{Q}}$, and Thomae's function $t$ with $t(x)=1\mathrm{/}q$ at a rational $x=p\mathrm{/}q$ in lowest terms with $q\ge 1$ and $t(x)=0$ at every irrational $x$).

A sequence of reals converges to $x$ when for every rational $\epsilon >0$ there is $K$ with $\mathrm{\mid }{x}_k-x\mathrm{\mid }<\epsilon$ for all $k\ge K$; an eventually constant sequence converges to that constant (Limits and Cauchy sequences of reals, Sequences of reals: bounded, eventually, frequently, tails, subsequences, Basic properties of the absolute value).

Every nonempty finite set of reals has a minimum (Every nonempty finite set of reals has a maximum and a minimum, Maximum and minimum of a set).

Continuity at a point: for every real $\epsilon >0$ there must be a real $\delta >0$ with $\mathrm{\mid }h(y)-h(x)\mathrm{\mid }<\epsilon$ for every $y$ in the domain with $\mathrm{\mid }y-x\mathrm{\mid }<\delta$ (Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point, The $\epsilon$-neighbourhood and the punctured $\epsilon$-neighbourhood of a point of $\mathbb{R}$).

Ordered-field arithmetic: the order is total and transitive, $\mathrm{\mid }u-v\mathrm{\mid }>0$ for $u\mathrm{\ne }v$, and $0<1$ (Basic properties of the absolute value, Order is preserved by adding a constant and by adding inequalities, Sign rules for products and monotonicity of multiplication, Ordered field, Complete ordered field (least-upper-bound property), Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length). These order-arithmetic facts are stated by their sources for the strict order only; the nonstrict forms used below follow by adjoining the equality case, in which the two sides coincide.