# How does every real number have a decimal representation?

I am a little bit confused at how every real number can have at least one decimal representation, because:

• There are uncountable many real numbers
• There are countable strings of infinite length over a finite alphabet

I would expect that there can be no injection from the real numbers to the infinite strings over {0,1,2,3,4,5,6,7,8,9,.}. Is there a mistake in my assumption or reasoning? Does this only mean that some different real numbers must have the same decimal representation?

• The second statement is false even for the alphabet {$0,1$} Cantor's diagonal method shows that there are uncountable many infinite long words. The set of finitely long words over some finite alphabet however is countable. Jul 31 at 15:16

You can see by diagonal argument that the set of infinite strings is uncountable: assume it is countable, and order the strings $$s_1,s_2,\ldots$$, then construct a string which is different to $$s_1$$ in position $$1$$, different to $$s_2$$ in position $$2$$, and so on.