There are uncountably many sentences of countably infinite length. The sentence at hand is countably long and thus cannot contain them all. It cannot even contain two countably infinite strings that don’t have a common infinite suffix.
There are uncountably many infinite strings, if you permit any content whatsoever. But when you have some restrictions, that may be, or may not be the case.
Take for example rational numbers a/b, where 0<=a<b and a and b are naturals. They represent countably infinite number of infinite sequences.
There are uncountably many sentences of countably infinite length. The sentence at hand is countably long and thus cannot contain them all. It cannot even contain two countably infinite strings that don’t have a common infinite suffix.
But you are right. To prove that there are actually uncountably many such infinite sequences are rather trivial.
So, you have solved the problem. Congratulations!
There are uncountably many infinite strings, if you permit any content whatsoever. But when you have some restrictions, that may be, or may not be the case.
Take for example rational numbers a/b, where 0<=a<b and a and b are naturals. They represent countably infinite number of infinite sequences.
These restrictions here may do the same.