Uncountable vs Countable Infinity

Your friend's method fails immediately for any number whose decimal expansion does not terminate. What is the index of $\pi $? For that matter, what is the index of $\frac13=0.333...$? In this sense, your friend's method fails for almost all rationals even.

(Migrated from the comments.)

Since you only have terminating decimals, you could just as well end each of them with an infinite string of zeroes. This does not change the value of any of the numbers that you had included, so if you had them all, then you will still have them all!

(For example: $0.31$ would become $0.31000\ldots$)

Once you have re-written the numbers as above, you can essentially apply Cantor's diagonal argument to these decimal representations to produce a number that is not found in your list.