Riemann integrals and step functions

[GridironCPJ]

Prove the following:

If f is Riemann integrable on an interval [a,b], show that ∀ε>0, there are a pair of step functions
L(x)≤f(x)≤U(x)
s.t.
∫_a^b▒(U(x)-L(x))dx<ε

My proof:

Since f is Riemann integrable on [a,b] then, by Theorem 8.16, ∀ε>0, there is at least one partition π of the interval [a,b] s.t.
∑_(k=1)^n▒〖ωf([x_(k-1),x_k ])(x_k-x_(k-1))〗<ε
Let L(x)=inf⁡〖f(x)〗 ∀x∈[x_(k-1),x_k] and let U(x)=sup⁡〖f(x)〗 ∀x∈[x_(k-1),x_k]. Note that
ωf([x_(k-1),x_k ])(x_k-x_(k-1) )=U(x)-L(x)
so

∑_(k=1)^n▒〖(U(x)-L(x))(x_k-x_(k-1))〗<ε
|∑_(k=1)^n▒〖(U(x)-L(x))(x_k-x_(k-1))〗-0|<ε
Hence,
∫_a^b▒(U(x)-L(x))dx<ε
Which is precisely the statement needed to be proven. ∎
I feel like my proof makes sense, but I would like to get some feedback to see if anyone sees any flaws in my logic. Note that I essentially explained the details of Theorem 8.16 in my proof. Also, ignore the ▒'s, as they were created when copying and pasting from Word.

 

[GridironCPJ]

I've attached a PDF of my proof, which should be much easier to read. The converse is also proved in the following problem.View attachment Step Functions and Riemann Integrals.pdf

Your critiques would be helpful!