Linearity of the Riemann Integral of Continuous Functions, and Passage to a Uniform Limit

In the setting of The Real Numbers: Standing Notation and Background, let $a,b$ be real numbers with $a<b$, let $[a,b]$ be the closed interval determined by $a$ and $b$, regarded as a subset of the real line $(\mathbb{R},d_{\mathbb{R}})$, and let the codomain $\mathbb{R}$ carry the same metric $d_{\mathbb{R}}$. Continuity on $[a,b]$ is that notion for maps from $[a,b]$ to $\mathbb{R}$ formed with these metrics, and $|t|$ denotes the absolute value of $t\in\mathbb{R}$. Every integral written below is the Riemann integral on $[a,b]$ of a function continuous on $[a,b]$, which is Riemann integrable there by A Continuous Function on a Closed Interval is Riemann Integrable §integrable. Pointwise sums and scalar multiples of functions on $[a,b]$ are formed as in The Real Vector Space of Real-Valued Functions on a Set. Uniform convergence of a sequence of functions and uniform convergence of a series of functions are as defined there, read with $D=S=[a,b]$, and convergence of a series of real numbers is as defined there. Then the following hold.

1. (Linearity)¶ Let $h_{1},h_{2}:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$ and let $c\in\mathbb{R}$. Then $h_{1}+h_{2}$ and $c\,h_{1}$ are continuous on $[a,b]$, and

2. (Order bound)¶ Let $h:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$ and let $M\in\mathbb{R}$ satisfy $|h(t)|\le M$ for every $t\in[a,b]$. Then

3. (Uniform limits)¶ Let $(h_{k})_{k\in\mathbb{N}}$ be a sequence in the set of functions from $[a,b]$ to $\mathbb{R}$, let $h:[a,b]\to\mathbb{R}$, suppose that $h_{k}$ is continuous on $[a,b]$ for every $k\in\mathbb{N}$, and suppose that $(h_{k})_{k\in\mathbb{N}}$ converges uniformly to $h$ on $[a,b]$. Then $h$ is continuous on $[a,b]$, and the sequence of real numbers $\bigl(\int_{a}^{b}h_{k}(t)\,dt\bigr)_{k\in\mathbb{N}}$ converges to $\int_{a}^{b}h(t)\,dt$.

4. (Termwise integration of a series)¶ Let $(g_{k})_{k\in\mathbb{N}}$ be a sequence in the set of functions from $[a,b]$ to $\mathbb{R}$, let $g:[a,b]\to\mathbb{R}$, suppose that $g_{k}$ is continuous on $[a,b]$ for every $k\in\mathbb{N}$, and suppose that the series $\sum_{k=1}^{\infty}g_{k}$ converges uniformly to $g$ on $[a,b]$. Then $g$ is continuous on $[a,b]$, the series $\sum_{k=1}^{\infty}\int_{a}^{b}g_{k}(t)\,dt$ converges, and