A difficulty in understanding the proof of Riemann Lebesgue lemma.

Question

The proof is given below:

My questions are:

1- In the second line from below how do we get $2/|\lambda|$ (in the second term) from the line before it.

2- What is the lemma trying to say?

Answer 1

For your first question, consider

$$ \max_{x_1,x_2} \left|\int_{x_1}^{x_2}\cos (x) dx \right|= \max_{x_1,x_2} \left|\sin (x) \big|_{x_1}^{x_2}\right| = 2.$$

That is, the maximum value of $\sin(x_2)-\sin(x_1)$ is 2. For an example, you can take $x_1=-\pi/2$ and $x_2=\pi/2$. The proof continues by using

$$\left|\sin (x) \big|_{x_1}^{x_2}\right| \le 2.$$

For the second question, the lemma itself says that the integral of an integrable function $F$ against a sinusoidal function will vanish in the limit of infinite oscillations over a finite interval. Basically, there are as many positive contributions as negative due to the infinite frequency, so the result is zero.