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'Weekly Challenge 11: Unit Interval' printed from http://nrich.maths.org/
Although this solution is quite short to write down, you do need to
keep a clear head to find it! As you read this proof, think
carefully about each step to be sure that you follow it.
Suppose that $0< x< 1$ and $0< y< 1$ for real numbers
Since $0< y< 1$ we must also have $0< 1-y< 1$.
Similarly, $0< 1-x< 1$.
Since the product of two real numbers between $0$ and $1$ must also
be between $0$ and $1$ we have
$$0< (1-x)(1-y)=1-x-y+xy< 1$$
Looking at the left hand side of this inequality we have
Rearranging gives the desired result.
Note: In this proof we assume standard properties of real numbers,
such as "the product of two real numbers between $0$ and $1$ must
also be between $0$ and $1$". You might wish to read this proof
carefully and try to note where assumptions such as these have been