I'm studying the Cantor diagonal argument showing that there isn't any function $f:\mathbb N \to \mathbb R$ in one-to-one with each other and accepted it. But, thinking about it for awhile, I thought in a function $f:\mathbb N \to \mathbb N$ in binary base, that is just the identity function: $$f(1) = 1$$ $$f(10) = 10$$ $$f(11) = 11$$ $$f(100) = 100$$ $$f(101) = 101$$ $$\dots$$ So, what's wrong if I define the following function: $$f(1) = 0.1$$ $$f(10) = 0.01$$ $$f(11) = 0.11$$ $$f(100) = 0.001$$ $$f(101) = 0.101$$ $$\dots$$
Where the only difference is that the digits are written in reverse order. Why the latter defined $f$ doesn't maps $\mathbb N$ in the real interval $(0,1]$?
