Understanding the Inequality in Trigonometric Function Analysis

[gajohnson]

Homework Statement

This isn't really a homework question, just working through Rudin and got caught up on something. $C(x)$ and $S(x)$ refer to $cos(x)$ and $sin(x)$ respectively.

Here is the section in question:

http://grab.by/mSo8

Homework Equations

The Attempt at a Solution

Well the part I'm having trouble understanding is the claim: "Hence, if $0≤x≤y$, we have $S(x)(y-x)<\int^{y}_{x}{S(t)}dt = C(x)-C(y)≤2$"

In particular, the inequality $S(x)(y-x)<\int^{y}_{x}{S(t)}dt$ is not clear to me. I reviewed a number of integration theorems but couldn't come up with anything that states this. Any help understanding how this inequality is derived would be much appreciated!

EDIT: OK, this might be really obvious. Is this simply true by the definition of the Riemann integral?

 

[haruspex]

It follows from the observation that S(t) is strictly increasing on the interval, so within the interval S(x) < S(t). Then integrate both sides over the interval.

 

[gajohnson]

Ah, of course. Got it. Thanks!