MathDB

Let

n \geqslant 3

be a positive integer and let

\left(a_{1}, a_{2}, \ldots, a_{n}\right)

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

\left(b_{1}, b_{2}, \ldots, b_{n}\right)

with sum equal to 2 such that

\sum_{i \in X} b_{i}=1.

A3