For various reasons, mainly of convenience. One good reason is given by considering the embedding of the rationals in the reals; the decimal approximation of $1$ for example looks a bit silly - one could conceivably write $0.9,0.99,0.999,\cdots$ but otherwise one obtains a finite sequence, which looks strange. Another reason is simply to avoid tying the construction to the decimal counting system. The usual construction is completely unambiguous, more consistent, and has nice operations.
Speaking of the operations - what else would you define addition as? Clearly you need to sum infinitely many pairs because the result must be an infinite set, and there is no natural way to choose only certain pairs in the sets $x$ and $y$ when computing $x+y$. Indeed, if the resulting set must contain all rationals below some cut, you need to consider many possible additions or you may miss some.
Think about multiplication for your sequence now - how would you define $\sqrt 2\times \sqrt 2$? You don't get a sequence with one more digit every time just by multiplying together a particular series of elements. Therefore you need more manipulation to fix this to be of the correct form. This is a big pain!
Are not the terminating decimals sufficient ?
Yes, you could also define things only in terms of terminating decimals, but again there is work to do, and since termination depends on which base you work in this isn't very natural. (It's tied to base 10 again and for no good reason; we're already familiar with non terminating expressions like $1/3$.)
The thing this makes annoying is division of reals. Dividing two terminating rationals doesn't always give you another way ($\frac 1 3$ again) so this requires special attention.
Basically, the point is that the construction should encompass the (ordered) field properties of $\mathbb Q$ if it is to be "nice". Terminating decimals are not a field.