Theorem 1.4. of Serge Lang´s Undergraduate Analysis

Question

Could somebody explain to me the boxed inequality. I have the feeling that the second member should be $L_1/2^{K-1}$. Thanks.

Answer 1

Yes, this looks like an error to me.

By construction, for $k,m\geq K, \space x_{n_k}$ and $x_{n_m}$ are both in $I_K$ which has length $L_K=\frac{L_1}{2^{K-1}}$. Thus, $|x_{n_k}-x_{n_m}|\leq \frac{L_1}{2^{K-1}}$.