Question :
Show that $f : [a,b]\to \mathbb R$ is Riemann integrable on $[a,b]$ if and only if there exists $L\in \mathbb R$ such that for every $\epsilon \gt 0$ there exists $\delta_\epsilon \gt 0$ such that if $\dot{\mathcal P}$ is any tagged partition with norm $||\dot{\mathcal P}|| \leq \delta_\epsilon$, then $|S(f;\dot{\mathcal P}) - L|\leq \epsilon$.
This question is from Bartle's book "Introduction to Real Analysis Fourth Edition" (Exercise 7.1 No.3).
I'm totally lost. My lecturer didn't explain enough about Riemann Integral and didn't give me some work example so I really have no idea of what I should do first or where to start. One thing that I notice is even though the asked question is identical with the Riemann Integral definition, in this question it was given "if $||\dot{\mathcal P}|| \leq \delta_\epsilon$, then $|S(f;\dot{\mathcal P}) - L|\leq \epsilon$" where the inequality sign is different from the definition.
I know from the definition, we need to confirm if $||\dot{\mathcal P}|| \lt \delta_\epsilon$, then $|S(f;\dot{\mathcal P}) - L|\lt \epsilon$, but how do I begin?
Thanks in advance!