I need a little confirmation on this one, it seemed a bit too easy for me, which makes me think I assumed too much:

Let T be a linear operator on an inner product space V, suppose ||T(x)||=||x|| for all x in V. Prove T is injective.

Here is my proof:

let x,y exist in V such that T(x)=T(y).

So ||T(x)||=||x|| implies

||T(x)||=||T(y)|| implies

||x||=||y|| implies

x=y. (Its this implication Im suspicious of).

Any help please?