I get the idea of how to do this proof, essentially I would choose a pathological partition $P$ such that it includes a point $ c \in P $ where $ f(c) \Delta_c - l \geq \epsilon$. However I'm stuck on how to make this work in a $ \delta,\epsilon $ proof.
Suppose the contrary that $ f $ is Riemann Integrable but $ f $ is not bounded. It suffices I show
$$ \exists \epsilon > 0, \forall \delta > 0\ \text{s.t}\ \exists P, m(P) < \delta\ \mathrm{but}\ |R(f,P) - l| \geq \epsilon $$
- $ m(P) $ is the mesh of the partition $P$
Since $ f $ is not bounded, choose $ c $ such that $ f(c) > \frac{l + \epsilon}{\delta} $ and let $ c \in P $
Then I initially I had this:
$$ |R(f,P) - l| = \left|\sum_{{s_j} \in P} f(s_j) \Delta_j - l\right| $$ $$ \geq \left|\sum_{{s_j} \in P} f(s_j) \delta - l\right|$$ $$ \geq \left|f(c) \delta - l\right|$$ $$ \geq \left|\frac{l + \epsilon}{\delta} \delta - l\right|$$ $$ = \epsilon $$
However, clearly the inequality here is nowhere near correct, since the difference is an absolute value. How would I go about this proof?
(I'm following the Stephen Krantz, Real Analysis and Foundations Textbook)
