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Weekly Challenge 11: Unit Interval

Stage: 5 Short Challenge Level: Challenge Level:1

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 $x, y$.
 
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
 
$$
0< 1-x-y+xy
$$
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 made.