Question 68
Let $n \geqslant 3$ be a positive integer and let $(a_{1}, a_{2}, \ldots, a_{n})$ be a strictly increasing sequence of $n$ positive real numbers with sum equal to $2$. Let $X$ be a subset of $\{1,2, \ldots, n\}$ such that the value of
$$
\left|1-\sum_{i \in X} a_{i}\right|
$$
is minimised. Prove that there exists a strictly increasing sequence of $n$ positive real numbers $(b_{1}, b_{2}, \ldots, b_{n})$ with sum equal to $2$ such that
$$
\sum_{i \in X} b_{i}=1
$$
$$
\left|1-\sum_{i \in X} a_{i}\right|
$$
is minimised. Prove that there exists a strictly increasing sequence of $n$ positive real numbers $(b_{1}, b_{2}, \ldots, b_{n})$ with sum equal to $2$ such that
$$
\sum_{i \in X} b_{i}=1
$$