A rather simple question I suppose, however I have not seen a discussion on this topic on stackexchange as of yet. In fact, when I was in high school I once used the following 'proof' to prove to my teacher that $1.9999... = 2$, where the dots indicate an 'infinite' number of decimal $9$'s:
$$x = 1.99999...$$
$$10x = 19.9999...$$
$$9x = 19.9999... - 1.9999... = 18$$
$$x = 2$$
$$1.9999... = 2$$
My teacher, who has a Masters degree in Mathematics in university, then told me that infinity is a tricky concept in mathematics and that this needn't work the way it seems to. Now, many many years later, being an econometrics student myself, I have found myself thinking about this 'proof'. To me, it seems that claiming a number to have an 'infinite' number of decimal $9$'s is faulty in itself; it would be much more logical to state that the number of $9$'s approaches infinity and as a result, that $1.9999...$ is asymptotically equal to $2$, but not equal to $2$.
However, when Googling this question I am overwhelmed with the number of results where people claim that it is in fact exactly, literally, equal: that $1.9999... = 2$ is undisputably true. As a result I am now unsure whether it is or is not true.
Thus, is $1.9999... = 2$?
