In modern mathematics, the string of symbols 
is understood to
be a shorthand for ``the infinite sum 
''.
This in turn is shorthand for ``the limit of the sequence of real numbers
9/10, 9/10 + 9/100, 
''. Using the
well-known epsilon-delta definition of the limit (you can find it in any of
the given references on analysis), one can easily show that this limit is 1.
The statement that 
is simply an abbreviation of this fact.
Choose 
. Suppose 
, thus
. For every 
we have that
So by the 
definition of the limit we have
Not formal enough? In that case you need to go back to the construction of
the number system. After you have constructed the reals (Cauchy sequences are
well suited for this case, see [Shapiro75]), you can indeed verify that the
preceding proof correctly shows .
An informal argument could be given by noticing that the following sequence
of ``natural'' operations has as a consequence . Therefore
it's ``natural'' to assume .
Thus .
An even easier argument multiplies both sides of 
by 3.
The result is 
.
Another informal argument is to notice that all periodic numbers
such as 
are equal to the period divided over the
same number of 9s. Thus 
. Applying the
same argument to 
.
Although the three informal arguments might convince you that , they are not complete proofs. Basically, you need to prove that each step
on the way is allowed and is correct. They are also ``clumsy'' ways to prove
the equality since they go around the bush: proving
directly is much easier.
You can even have that while you are proving it the ``clumsy'' way, you get
proof of the result in another way. For instance, in the first argument the
first step is showing that is real indeed. You can do this by
giving the formal proof stated in the beginning of this FAQ question. But
then you have as corollary. So the rest of the argument is
irrelevant: you already proved what you wanted to prove.
References
R.V. Churchill and J.W. Brown. Complex Variables and Applications. 
ed., McGraw-Hill, 1990.
E. Hewitt and K. Stromberg. Real and Abstract Analysis. Springer-Verlag, Berlin, 1965.
W. Rudin. Principles of Mathematical Analysis. McGraw-Hill, 1976.
L. Shapiro. Introduction to Abstract Algebra. McGraw-Hill, 1975.