Question 83
A real sequence $a_{1}, \ldots, a_{k}$ is said to fit in the interval $[b, c]$ if there exist real numbers $x_{0}, \ldots, x_{k}$ in $[b, c]$ such that $\left|x_{i}-x_{i-1}\right|=a_{i}$ for $k \geqslant i \geqslant 1$. The sequence is said to be normalised if all its terms are less than or equal to 1.
1) Prove that for every integer $n \geqslant 1$, every normalised sequence of length $2 n+1$ fits in $[0, 2-\frac{1}{2^{n}}]$.
2) Prove that for every integer $n \geqslant 1$, there exists a normalised sequence of length $4 n+3$ which does not fit in $[0,2-\frac{1}{2^{n}}]$.
1) Prove that for every integer $n \geqslant 1$, every normalised sequence of length $2 n+1$ fits in $[0, 2-\frac{1}{2^{n}}]$.
2) Prove that for every integer $n \geqslant 1$, there exists a normalised sequence of length $4 n+3$ which does not fit in $[0,2-\frac{1}{2^{n}}]$.