# Interactive Real Analysis

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## 7.1. Riemann Integral

### Examples 7.1.2(c):

Show that if P' is a refinement of P then | P' | | P |
To prove this fact is more confusing than enlightening. It seems clear that if one or more points are inserted into the partition P to form the refinement partition P', the largest distance between the points of P' must now be less than (or equal to) that of the points of P.

But alas, even things that "seem clear" still need formal proof, so ...

Since | P | is a maximum, there must be at least one integer j such that | P | = xj+1 - xj. Take all such points from the partition P, i.e. all points such that | P | = xj+1 - xj. Now consider the refinement P'.

• Suppose none of the additional points are inside the intervals [xj, xj+1]. Then the original maximum has not changed so that | P | = | P' |

• Suppose at least one of the additional points is inside at least one of the subintervals [xj, xj+1]. Then this subinterval can no longer contribute to the maximum of P' so that | P' | | P |
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