Assume that x and y are positive integers, prove that if x divides y then x<=y.

If x divides y then there exists another integer j such that y=xj.

Now we can say x =< xj and canceling out the x we get 1<=j or that is to say, j is greater than or equal to 1.

This is true because both x and y are positive so j must be positive for our initial antecedent y=xj to be true. Also since we assumed j is an integer then j>=1. So now we know that x<=xj and by substitution of y=xj we get x<=y.

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