Thread: Darboux Sums

So we were doing Darboux sums in Analysis, and this was a given sample question and answer, but I don't really understand where part of this solution comes from:

"Suppose f is increasing on [0,1]. Show that f is integrable on [0,1]

Let P = be a partition of [0,1] into n subintervals of the same length.
Since f(x) is increasing, we have

and this implies that





But I don't understand how those last two statements are equal; for example, how is





equal to

Originally Posted by Conn

So we were doing Darboux sums in Analysis, and this was a given sample question and answer, but I don't really understand where part of this solution comes from:

"Suppose f is increasing on [0,1]. Show that f is integrable on [0,1]

Let P = be a partition of [0,1] into n subintervals of the same length.
Since f(x) is increasing, we have

and this implies that





But I don't understand how those last two statements are equal; for example, how is





equal to

It follows from the fact that



Then just factor it out of each term!