Proof of integrability of a function when mesh(P)The following is a question that I got in class. Apparently it's an easy proof, and I have written one out, just need it checked.
Question:
Let $f$ : $[a,b]$ $\to \mathbb{R}$  and let $I \in \mathbb{R}$. Suppose that for every $\varepsilon \gt 0$, there exists some $\delta \gt 0$ such that:
$$\vert I - R(f;P,\xi)\vert \lt \varepsilon$$ 
for every partition $P$ of $[a,b]$ with mesh($P$) < $\delta$ and every choice of tag $\boldsymbol \xi $ of P. Prove that $f$ is integrable over $[a,b]$ and:
$$\int_a^bf(x) dx = I $$
Proof:
Let $\varepsilon \gt 0 $ arbitrarily. Let $P_1$ be a partition of $[a,b]$ which has mesh$(P_1) \lt \delta$. Now add a point to $P_1$ and we get the partition $P_2$ of $[a,b]$. Now, mesh$(P_2) \le $ mesh$(P_1) \lt \delta$. We can then add a point to the partition $P_2$ and we get $P_3$, a partition of $[a,b]$. Again, we have that mesh$(P_3) \le $ mesh$(P_2) \le $ mesh $(P_1) \le \delta$. Continuing this process indefinitely, we can form a sequence of partitions $(P_k)_{k=1}^\infty$ where mesh$(P_k) \le \delta$ for all $k$ and $\lim_{k \to \infty}$  mesh$(P_k)=0$. Now, we can write $\vert I - R(f;P,\xi)\vert \lt \varepsilon$ for all $k$ and so, taking limits:
$$\lim_{k \to \infty}\vert I - R(f;P,\xi)\vert \lt \varepsilon$$
$$\vert I - \lim_{k \to \infty}R(f;P,\xi)\vert \lt \varepsilon$$
by continuity of the Euclidean norm. So this (and the fact that $\varepsilon \gt 0$) implies that $\lim_{k \to \infty}R(f;P,\xi) = I$, as required. 
 A: Let $P_{k}$ be any sequence of partitions of $[a, b]$ such that $\mu(P_{k}) \to 0$ as $k\to\infty$. I have used $\mu(P)$ to denote mesh of partition $P$. Also let $\xi_{k}$ denote any choice of tagging for partitions $P_{k}$.
We are given that for every $\epsilon > 0$ there is a $\delta > 0$ such that $$|I - R(f;P,\xi)| < \epsilon$$ for all partitions $P$ and tagging $\xi$ of $P$ with $\mu(P) < \delta$.
We have to use the above condition to establish the following:
For any given $\epsilon > 0$ there is an integer $N$ such that $$|I - R(f;P_{k}, \xi_{k})| < \epsilon$$ for all $k \geq N$.
Now this is easy to do. For the given $\epsilon > 0$ we choose $\delta > 0$ (this is possible because of the given condition). Now $\mu(P_{k}) \to 0$ and hence there is a positive integer $N$ such that $\mu(P_{k}) < \delta$ whenever $k \geq N$. And then by given condition we have $$|I - R(f;P_{k},\xi_{k})| < \epsilon$$ whenever $k \geq N$. Thus we get the conclusion as desired and $f$ is integrable on $[a, b]$ with integral $I$.
