Talk:Real Numbers are Uncountably Infinite/Cantor's Diagonal Argument

"why is it a problem, that it needs overcoming?"

Because there is actually the tacit attempted equation of $[\N \to \{0\ldots 9\}]$ (mappings from $\N$ to $\N_{10} := \{0\ldots9\}$) and $[0\,.\,.\,1]$.

Since this is not actually a bijection (putting $x = 0.d_{x1}d_{x2}\cdots$ does not uniquely determine the $d_{xi}$) one has to restrict $[\N \to \N_{10}\}]$ to the slightly smaller collection mentioned on the page (i.e. $\{f \in [\N\to\N_{10}]: \forall M \in \N: \exists m \ge M: f(m)\ne 9\}$). Interesting how such an intuitively small, innocent statement can actually require quite a bit of work to explain properly. --Lord_Farin (talk) 13:02, 21 September 2012 (UTC)