Existence of the natural density of the strictly-increasing sequence of positive integer?

Consider the sequence of integers with an odd number of decimal digits.

Yes. Start with $1$. Then omit enough integers to reduce the ratio below $\frac{1}2$. Then include enough consecutive integers to increase the ratio above $1-\frac{1}3$. Then omit enough to reduce it below $\frac{1}4$. Then include enough to increase it above $1-\frac{1}5$. Keep going.