A rather convincing argument that they are "equal" is the method of finding the fraction from any repeating decimal. Let's review that.
What fraction is equal to 0.33333 .... ?
Step 1
Set 0.33333 .... equal to 'X'
X = 0.33333 ....
Step 2
Multiply by ten
10X = 3.33333 ....
Step 3
Subtract Step1 from Step 2
9X = 3
Finally, solve for X
X = 3/9, or reducing the fraction, 1/3
Using 0.99999 .... obtains 1 in a similar manner.
(Although we only multiplied by 10 once in these cases, it might be necessary to keep multiplying by ten until the the left of the decimal is not zero and the right of the decimal matches the right of the decimal before the multiplication began.)
What confuses people is that most schools often make the practical assumption that 0.33333 .... "means" 1/3 and that 0.99999 .... "means" one. They can do that because expressions can "mean" whatever we want them to mean. "When I say this I 'mean' that" is perfectly legal practice.
While the assumption of equality is very practical in most real world problems it is not truly an equality though. If you start with 0.3 and add a new digit every minute; 0.3, 0.33, 0.333, 0.3333, when will your result be "equal" to 1/3? In ten years? A thousand? The correct answer that it will never be truly "equal" to three. At this point some will complain that it becomes equal "in infinity." That is not correct because infinity has no limit. In calculus it is said that the value of the sum of 3 × 10-n "approaches" 1/3 as n goes from 1 to infinity. The value of 1/3 is called the "limit" of the sum and used in calculations. However this is merely a matter of defining things in a convenient way. Infinity has no limit. It is necessary to be mindful of the word problems as before.
What went wrong in the method of finding the equivalent fraction is the algebraic manipulation of infinite properties, which is not "legal" in mathematics. Infinity plus twelve or infinity times seven cannot really be algebraically manipulated. They make no sense. You will hear the expression of a "one to one correspondence" that is deemed "fair" in the method of finding the fraction, and it is generally accepted, however no algebraic manipulation of infinite properties is dependable. Notice how the correct answer to the question of at what time adding the digit 3 to an expression would "equal" one third could be lost.
If you really could allow infinite properties to be algebraically manipulated then you could speak of 1 divided by infinity.
Then the repeating decimal 0.99999 ... would equal 1-(1/∞).
The probable cause of most difficulties associated with the concept of infinity is the tendency to consider infinity a "quantity" or a "number." It is no such thing. That is the reason infinity cannot be used in algebraic manipulations. Such manipulations deal with quantities and infinity is not a quantity. The assumption that describing some "quantity" as infinite has somehow "fixed" the problem is false. There is no fixed quantity.
It can be especially difficult for some people to understand how something boundless can fit inside something that has bounds. For example a line segment of fixed length is bounded, it is finite. The number of points on that line segment is not bounded in the same sense. It is in fact more correct to say that there is no "number" of points. It is not possible to count all the points on the line. For any two points found there will be points between them, and points betweeen these, and so on, endlessly. This is possible because the points have no quantity themselves. They have no length, no volume, no quantity at all.
The trouble with the concept of "one third" of some unit of measure in the original article is the result of using something infinite to describe something finite. It can be problematic to describe something which is a quantity using something which is not a quantity. One third is indeed a finite quantity. The decimal description of it is not finite, just as the number of points on a finite line is not finite. That is the reason care is required in setting up the word problem. It is necessary to be clear which things end and which things do not end, which things are quantities and which not.
Attempting to describe the "result" of an ever continuing process can be more or less hazardous depending on circumstances. The assumption of "equality" needs to be justified.