Question 95
We say that a sequence of nonnegative real numbers $a_{1}, a_{2}, \ldots, a_{k}$ is embeddable in the interval $[b, c]$ if there exist numbers $x_{0}, x_{1}, \ldots, x_{k}$ from the interval $[b, c]$ such that $\left|x_{i}-x_{i-1}\right|=a_{i}$ for $i=1,2, \ldots, k$. A sequence is normalized if all its terms are not greater than 1. For a given natural number $n$, prove:
a) every normalized sequence of length $2 n+1$ is embeddable in the interval $\left[0,2-\frac{1}{2^{n}}\right]$;
b) there exists a normalized sequence of length $4 n+3$ which is not embeddable in $\left[0,2-\frac{1}{2^{n}}\right]$.
a) every normalized sequence of length $2 n+1$ is embeddable in the interval $\left[0,2-\frac{1}{2^{n}}\right]$;
b) there exists a normalized sequence of length $4 n+3$ which is not embeddable in $\left[0,2-\frac{1}{2^{n}}\right]$.