I have seen repeating numbers shown in all of the following ways:
$$1.333\dots$$
$$1.\overline{3}$$
$$1.\overline{3}\dots$$
$$1.\overline{33}$$
$$1.\overline{33}\dots$$
$$1.\overline{333}$$
$$1.\overline{333}\dots$$
The ones using ellipses feel incorrect, especially when paired with the line over the $3$. What is the proper convention for showing repeating numbers (in this case $1\frac{1}{3}$)?
I don't think ellipsis should be used if there is any other notation, because it isn't necessary.
The best ways for $1\frac 13$ are:
$1\frac 13 = 1.333...$ (I think that two digits after the decimal point might be sufficient)
$1\frac 13 = 1.\dot 3$
$1\frac 13 = 1.\overline 3$
Where there are two repeated digits, we use:
$1\frac {23}{99} = 1.2323...$
$1\frac {23}{99} = 1.\dot 2\dot 3$
$1\frac {23}{99} = 1.\overline {23}$
Although it wouldn't be wrong to use two dots or a bar over two digits when the two repeated digits are not the same, it's not the most elegant way.
Where there are three or more repeated digits, we often only use a dot at the start and the end of the repeated digits:
$1.523162316...$
$1.523162316... = 1.5\dot 2 31\dot 6$
$1.523162316... = 1.5\overline {2316}$
Remember that the use of these symbols is only our attempt to find a way of representing the number because a decimal fraction is not a good way to represent many fractions.