real analysis: prove that f is injective

[really.smarty]

Consider the function f(N) --> R given by f(A) = .d1d2d3 . . . where the decimal expansion of f(A) is determined by the rule that di = 0 if i is not in A and di = 1 if i is in A. Prove that f is injective.

I am stuck on this i would appreciate any help. Thank you.

 

[pka]

Consider the function f(N) --> R given by f(A) = .d1d2d3 . . . where the decimal expansion of f(A) is determined by the rule that di = 0 if i is not in A and di = 1 if i is in A. Prove that f is injective.

Suppose that \(\displaystyle f(A)=f(B)\) we want to show that \(\displaystyle A=B\).
So \(\displaystyle f(A)=f(B)=0.d_1d_2d_3\cdots\).
If \(\displaystyle j\in A\) then \(\displaystyle d_j=1\) which means that \(\displaystyle j\in B\)
If \(\displaystyle j\not\in A\) then \(\displaystyle d_j=0\) which means that \(\displaystyle j\not\in B\).
That means that \(\displaystyle A= B\) WHY?
Are wew done?