## 7.1. Riemann Integral

### Proposition 7.1.7: Size of Riemann Sums

Suppose

*P = { x*is a partition of the closed interval_{0}, x_{1}, x_{2}, ..., x_{n}}*[a, b]*,*f*a bounded function defined on that interval. Then we have:- The lower sum is increasing with respect to refinements of
partitions, i.e.
*L(f, P') L(f, P)*for every refinement*P'*of the partition*P* - The upper sum is decreasing with respect to refinements of
partitions, i.e.
*U(f, P') U(f,P)*for every refinement*P'*of the partition*P* -
*L(f, P) R(f, P) U(f, P)*for every partition*P*

### Proof:

The last statement is simple to prove: take any partition*P = {x*. Then

_{0}, x_{1}, ..., x_{n}}whereinf{ f(x), x_{j-1}x x_{j}} f(t_{j}) sup{ f(x), x_{j-1}x x_{j}}

*t*is an arbitrary number in

_{j}*[x*and

_{j-1}, x_{j}]*j = 1, 2, ..., n*. That immediately implies that

L(f, P) R(f, P) U(f, P)

The other statements are somewhat trickier. Let's first find
out why they should be true. To make it simple, let's say that
*P = {a, b}* and *P' = {a, x _{0}, b}*.
Then

and the upper sum forU(f, P) = sup{ f(x), x [a, b] } × (b - a)

*P'*would be

Geometrically, the upper sum forU(f, P') = sup{ f(x), x [a, x_{0}] } × (x_{0}- a) + sup{ f(x), x [x_{0}, b] } × (b - x_{0})

*P*corresponds to one large rectangle, the one for

*P'*to two smaller rectangles, where the smaller rectangles fit into the larger one but do not cover it.

U(f, P) = 1.089 |
U(f, P') = 0.86 |

*U(f, P) > U(f, P')*.

Let's show this mathematically, in case one additional point
*t _{0}* is added to a particular subinterval

*[x*. Let:

_{j-1}, x_{j}]*c*be the_{j}*sup*of*f(x)*in the interval*[x*_{j-1}, x_{j}]*A*be the_{j}*sup*of*f(x)*in the interval*[x*_{j-1}, t_{0}]*B*be the_{j}*sup*of*f(x)*in the interval*[t*_{0}, x_{j}]

*c*and

_{j}A_{j}*c*so that

_{j}B_{j}That shows that ifc_{j}(x_{j}- x_{j-1}) = c_{j}(x_{j}- t_{0}+ t_{0}- x_{j-1}) = c_{j}(x_{j}- t_{0}) + c_{j}(t_{0}- x_{j-1})

B_{j}(x_{j}- t_{0}) + A_{j}(x_{0}- t_{j-1})

*P = {x*and

_{0}, ... x_{j-1}, x_{j}, ..., x_{n}}*P' = {x*then

_{0}, ... x_{j-1}, t_{0}, x_{j}, ..., x_{n}}*U(f, P) U(f, P')*.

The proof for a general refinement *P'* of *P*
uses the same idea plus some confusing indexing scheme. No more
details should be necessary.

The proof for the statement regarding the lower sum is analogous.