# Riemann integrals and step functions

1. Apr 16, 2012

### 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.

2. Apr 16, 2012