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'Unit Interval' printed from https://nrich.maths.org/
Given any two numbers between $0$ and $1$ you have to prove that
their sum is less than 1 plus their product; that is, given $0 <
x < 1$ and $0 < y < 1$, prove that $x + y < 1 + xy$.
Hyeyoun Chung, St Paul's Girls' School, and Andaleeb Ahmed,
Woodhouse Sixth Form College, London both produced nice solutions.
Consider $1-x$ and $1-y$. Since $0 < x < 1$ and $0 < y
< 1$ it follows that
$\begin{eqnarray} \\ (1 - x)(1 - y) & > & 0 \\ 1 - x - y
+ xy & > & 0 \\ 1 + xy & > & x + y.
\end{eqnarray}$
This is equivalent to $x + y < 1 + xy$.