This has been widely discussed. Here is one link.
Near the end of that page there is another link to Cantor controversy.
These are a good place to start.
I know the proof of the uncountability of the real numbers using the open subset and a diagonalization argument, however I was reading a book which briefly mentions it and it says that it does not take into account the non-uniqueness of decimal representation e.g. 0.999....=1 and I was wondering how we get round this? Do we simply state the in our list of the subset of the real's we do not allow decimals to be represented in this way? Sorry but I am quite confused by this.
Thanks for any help