Prove that if $\forall \epsilon > 0 \; \exists$ integrable functions $h,g$ at $[a,b]$ s.t $h\leq f \leq g $ and $\int \limits_{a}^{b}(g-f) < \epsilon$ $\implies f$ is integrable.
We've discussed both Riemann and Darboux integrals, and defined integral using Darboux sums. I've tried to use the definition that $f$ is integrable $\iff$ $\forall \epsilon > 0 \; \exists$ step funcions $\phi$ and $\psi$, $\phi \leq f \leq \psi$ s.t $\int \limits_a^b f-\phi < \epsilon$ and $\int \limits_a^b \psi-f < \epsilon$. However I didn't manage to define either $\phi$ or $\psi$ that will do the job. Any ideas?
Let $P=\{x_0=a,x_1.......x_n=b\}$ a partition of $[a,b]$
Take $\phi(x)=\sum_{k=1}^nl_k\mathbb{X}_{[x_{k-1},x_k]}(x)$ where $l_k=\inf\{f(x):x \in [x_{k-1},x_k]\}$
And $\psi(x)=\sum_{k=1}^nu_k\mathbb{X}_{[x_{k-1},x_k]}(x)$ where $u_k=\sup\{f(x):x \in [x_{k-1},x_k]\}$
These two functions are simple functions..
$\mathbb{X}_A$ is the indicator function of the set $A$
You can continue from here..