If $A$ is the set of all positive integers with an odd number of digits, let $A\,(N)$ be the number of elements of $A$ which are less than or equal to $N$. Show that $P\,(A) = \lim\_{N\to\infty} A\,(N)/N$ does not exist.

My rusty knowledge of limits says $P\,(A) = 0$, since the denominator dominates. Any help?