Fibonacci numeral system. Prove that any natural number $n$, not exceeding $F_{\mathrm{m}}$, can be uniquely represented in the form

$$n=\sum_{k=2}^{m} b_{k} F_{k},$$

where all numbers $b_{2}, \ldots, b_{\mathrm{m}}$ are either 0 or 1, and among these numbers there are no two consecutive ones, that is, $b_{\mathrm{k}} b_{\mathrm{k}}+1$ $=0(2 \leqslant k \leqslant m-1)$. For representing a number in the Fibonacci numeral system, the notation used is:

$$n=\left(b_{\mathrm{k}} \ldots b_{2}\right)_{\mathrm{F}}$$