Because in a proof by contradiction, you have to assume that the property is true, which is not what you did, since you said that $\displaystyle n\neq 1$. And you have to conclude with a contradiction.

See the truth table for $\displaystyle A\Rightarrow B$

If A is false, then B can be either false or true, the proposition "$\displaystyle A\Rightarrow B$" will always be true.

In order to prove the property, you can just use the property of addition of natural numbers :

If a and b are natural numbers, then a+b is also a natural number.

Hence 1+1 is a natural number. And 1+1>1. Thus the statement is false