Why is it true that the lower sums of f with respect to some partition is less than the lower integral (which is the supremum of the lower sums)

I think what I'm confused about is the difference between a lower sum and the lower integral, how can the lower integral be the supremum of the lower sums when a lower sum is a single number?

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    $\begingroup$ You basically answered your question one the first sentence. $\endgroup$ – Git Gud Jan 18 '15 at 21:28
  • $\begingroup$ It is referring to a supremum of a set of lower sums given all possible partitions, not just the partition in question. The answer to your question is given by the definition of what a supremum is. $\endgroup$ – Darren Jan 18 '15 at 21:51
  • $\begingroup$ @DarrenNaylor oh so lower sums refers to a set of lower sums and so then the lower integral would refer to the smallest number in the set? $\endgroup$ – Nicole Jan 18 '15 at 23:15
  • $\begingroup$ Not quite. The supremum is the least number that is greater or equal to all the numbers in the set. $\endgroup$ – Darren Jan 19 '15 at 0:17

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