Can every real number be represented by a (possibly infinite) decimal? Does every real number have a representation within our decimal system? The reason I ask is because, from beginning a mathematics undergraduate degree a lot of 'mathematical facts' I had previously assumed have been consistently modified, or altogether stripped away. I'm wondering if my subconscious assumption that every real number can be represented in such a way is in fact incorrect?
If so, is there a proof? If not, why not?
(Also I'm not quite sure how to tag this question?)

Yes, every real number has a decimal representation.
Numbers of the form $n/10^k$ where $n$ and $k$ are integers and $n \neq 0$ have two representations each (e.g. 1.000... and 0.999... represent the same real number). Depending on your conventions, zero has either one or two representations: e.g. $0$ and $-0$ in the latter case. Every other real number has only one decimal representation.
Conversely, every decimal number that is all zeroes on the left is the representation of some real number. e.g. the decimal
$$ ...000011.000... $$
is the number eleven. (recall that any digit that isn't written is zero, so the decimal "11" really has zeroes to its left and right, as notated explicitly above)
In standard notation, decimals that are not all zeroes on the left such as
$$ ...11111.000... $$
do not represent real numbers.

There are generalized notations that allow some left-infinite decimals to represent real numbers (the number above would be -1/9). There are also other number systems that can be represented with decimals, such as the 10-adic numbers. It may be confusing to try and learn about such things at this point, though....
