I think this goes in this section.
Let x be a real number. Prove that
abs(sin nx) <= n*abs(sinx) for all positive integers n.
Mathematical induction is the tool here.
It is true for thus, there is a such as,
Thus,
But,
Thus, (mutiply inequalities notice they are non-negative),
......(1)
Also,
And,
Thus, (mutiply inequalites notice they are non-negative),
......(2)
Using the property on (1) and (2) we have,
Now, add these inequalities,
Now, by triangular inequality,
By transitivity ( ),
Thus, recognizing the sum for sine and simply the right side,
Proof is complete.