Proof of the starting part of theorem 1.17 Rudin ( Complex and Reals)

Question

The proof I would like is of the following fact:

Put $\delta_n = 2^{-n}$. To each positive integer n and each real number t corresponds a unique integer $ k = k_n(t)$ that satisfies $k \delta_n \le t < (k+1) \delta_n$ .

This is the start of the theorem 1.17 Rudin.

Answer 1

This comes from the Archimedean property: let $k+1$ be the least integer larger than $t/\delta_n$.