# Prove that Upper Riemann Integral is greater than or equal to lower Riemann integral

## Proof;

To prove that lower Riemann integral is less than or equal to upper Riemann integral or Upper Riemann Integral is greater than or equal to lower Riemann integral we have to show that supremum of set of lower Riemann sum is less than or equal to infimum of set of upper Riemann sum.

As

Lower Riemann integral = Sup {L(P , f) :P€S}

Upper Riemann integral = inf {U(P , f) :P€S }

Where S denotes the set of partitions.

As we know that for any arbitrary partitions p and Q we have

L(P , f) ≤ U(Q , f)

Since

{ L(P , f) : P€S } ≤ U(Q , f)

This implies that

Sup { L(P , f) : P€S } ≤ U(Q , f)

This implies that

Lower Riemann integral ≤ U(Q , f)

And now as

Lower Riemann integral ≤ U(Q , f)

This implies that

Lower Riemann integral ≤ inf { U(Q , f) :Q€S}

This implies that

Lower Riemann integral ≤ Upper Riemann Integral

Lower Riemann integral is less than or equal to upper Riemann integral |

Upper Riemann integral is greater than or equal to lower Riemann integral |