I looked at your enumeration of the continuum and it appears that you have enumerated only the terminating decimals (equivalently decimals that end in infinite trailing zeros or 9's). You have not enumerated all of the real numbers. If we focus on the enumeration of [0, 1) , how does your list produce the real number 1/3 = 0.333...? This real number is not a terminating decimal (nor does it end in infinite trailing zeros or nines). It will never be in the list - even if you are given an infinite amount of time to list numbers. Said another way, the real number 1/3 is not in the 'eventually-we-will-get-there' list, since your enumeration or listing method has no way of producing 0.333..., it is not in the list at all. A similar argument can be made for √2, but I will use 1/3 as a simpler counterexample.
I looked at your enumeration of the continuum and it appears that you have enumerated only the terminating decimals (equivalently decimals that end in infinite trailing zeros or 9's). You have not enumerated all of the real numbers. If we focus on the enumeration of [0, 1) , how does your list produce the real number 1/3 = 0.333...? This real number is not a terminating decimal (nor does it end in infinite trailing zeros or nines). It will never be in the list - even if you are given an infinite amount of time to list numbers. Said another way, the real number 1/3 is not in the 'eventually-we-will-get-there' list, since your enumeration or listing method has no way of producing 0.333..., it is not in the list at all. A similar argument can be made for √2, but I will use 1/3 as a simpler counterexample.