Prove if x*y = 1 then x = y = 1

Assume that x and y are positive integers, prove that if x * y = 1 then x = y = 1

If x * y = 1 then x = (1/y) and y = (1/x)

Yesterday we proved that if an integer x divides an integer y then x<=y.

So since y divides 1 or (1/y) then y<=1

and since x divides 1 or (1/x) then x<=1

Since both x and y are integers then y=1 and x=1, so x = y = 1