Let $I=[a,b]\subset\mathbb{R}$ be a closed and bounded intervall and let $f:I\to\mathbb{R}$ be a bounded function. For a subdivision $\sigma$ of $I$ (i.e. a finite strictly increasing sequence $(x_0,...,x_N)$ with $x_0=a$ and $x_N=b$) we define $$ \overline{S}_{\sigma}(f)=\sum_{i=1}^{N}(x_i-x_{i-1})\sup_{x\in[x_{i-1},x_i]}f(x)\\ \underline{S}_{\sigma}(f)=\sum_{i=1}^{N}(x_i-x_{i-1})\inf_{x\in[x_{i-1},x_i]}f(x) $$ and thereby $$ \overline{S}(f)=\inf\{\overline{S}_\sigma(f)\ :\ \sigma\text{ subdivision of } I\}\\ \underline{S}(f)=\sup\{\underline{S}_\sigma(f)\ :\ \sigma\text{ subdivision of } I\} $$ Using this, we define the space of Riemann integrable functions over $I$, denoted by $\mathcal{R}(I)$, as $$ \mathcal{R}(I)=\{f:I\to\mathbb{R}\ :\ f \text{ bounded},\ \overline{S}(f)=\underline{S}(f)\} $$ and for $f\in\mathcal{R}(I)$ $$ \int_I f=\overline{S}(f). $$ I have three questions:
(1) If $\|\cdot\|$ denotes the uniform function norm over $I$ (i.e. $\|f\|=\sup_{x\in I} f(x)$), then is it true that $(\mathcal{R}(I),\|\cdot\|)$ is complete?
(2) Is it equivalent to the above definition when we consider only the uniform subdivisions $\tau_{N}=(x_0,...,x_N)$ with $x_i=a+i\frac{b-a}{N}$? More precisely, is it always true that $$ \overline{S}(f)=\inf\{\overline{S}_{\tau_N}(f)\ :\ N\in\mathbb{N}\}\\ \underline{S}(f)=\sup\{\underline{S}_{\tau_N}(f)\ :\ N\in\mathbb{N}\}? $$
(3) For a subdivision $\sigma=(x_0,...,x_N)$ of $I$ we define $\Delta(\sigma)=\max_{i=1}^{N}x_i-x_{i-1}$. Is it true that for a Riemann integrable function $f$ and a sequence of subdivisions $\{\sigma_n\}_{n\in\mathbb{N}}$ with $\lim_{n\to\infty}\Delta(\sigma_n)=0$ we have $$ \int_I f=\lim_{n\to\infty}\overline{S}_{\sigma_n}(f)=\lim_{n\to\infty}\underline{S}_{\sigma_n}(f)? $$ This is true if $f$ is continuous (note that $C^0(I)\subset\mathcal{R}(I)$), however I fail to see if it is true for non-continuous functions.
How to prove the answers to the three questions? Questions (2) and (3) are somewhat related, although not completely equivalent. If you think it would be better to create one post for one question each, then leave a comment and I will change it.