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Given that 0<x<1, 0 <y<1.
We have to prove that x+y < 1+xy
Claim: xy < x, and xy < y
proof: Let xy < x is not true i.e. xy >= x. Since x is not zero. we can multiply inverse of x both sides. This implies y>=1. This is contradiction to our assumption. Similarly, we can prove xy < y.
Now, we prove x+y < 1+xy
Let this is not true i.e. x+y >= 1+xy. This implies x(1-y) >= 1-y. From the above claim, we have 1-y <=x(1-y)<1-y i.e. 1-y < 1-y. This is contradiction to our assumption.
Thus, x+y < 1+xy.