RS-SMO 2019 Problem 2

For the sequence of real numbers $a_1,a_2,\dots ,a_k$ we say it is invested on the interval $[b,c]$ if there exists numbers $x_0,x_1,\dots ,x_k$ in the interval $[b,c]$ such that $|x_i-x_{i-1}|=a_i$ for $i=1,2,3,\dots k$ . A sequence is normed if all its members are not greater than $1$ . For a given natural $n$ , prove :

a)Every normed sequence of length $2n+1$ is invested in the interval $\left[ 0, 2-\frac{1}{2^n} \right ]$ . b) there exists normed sequence of length $4n+3$ wich is not invested on $\left[ 0, 2-\frac{1}{2^n} \right ]$ .