A difficulty in understanding the proof of Riemann Lebesgue lemma(2)

Question

A part of the proof is below:

1-But it seems to me that the last line the last inequality is incorrect ….. am I correct?

2-Also if I am going to prove that the second limit equals zero I will use $$ \max_{x_1,x_2} \left|\int_{x_1}^{x_2}\sin (x) dx \right|= \max_{x_1,x_2} \left|-\cos (x) \big|_{x_1}^{x_2}\right| = 2.$$ am I correct?

I think this link may help:

A difficulty in understanding the proof of Riemann Lebesgue lemma.

Answer 1

You are right.

It should be

• $L= 4\sum_{i=1}^n\frac{|m_i|}{\epsilon}$ and then consider $\lambda > L$.

The other part is just triangle inequality:

• $\left|\sin \lambda t\right|_{t_{i-1}}^{t_i} \leq |\sin \lambda t_{i-1}| + |\sin \lambda t_{i}| \leq 2$