Mistake in proof of Lemma 2.3 in Chapter 3 of Stein and Shakarchi's Fourier Analysis
Solution 1:
Note that $1-r=1-(1-1/N)=1/N$. Then:
$$MN(1-r)+\frac{M}{N}\frac{1}{1-r}=MN(1/N)+\frac{M}{N}\frac{1}{1/N}=M+\frac{M}{N}\cdot N=2M$$
Mistake in proof of Lemma 2.3 in Chapter 3 of Stein and Shakarchi's Fourier Analysis
Solution 1:
Note that $1-r=1-(1-1/N)=1/N$. Then:
$$MN(1-r)+\frac{M}{N}\frac{1}{1-r}=MN(1/N)+\frac{M}{N}\frac{1}{1/N}=M+\frac{M}{N}\cdot N=2M$$