Cardinality of a set of natural sequences

Question

Let $a=(a_n)_{n\ge 1}$ a sequence such that for every $n\ge 1$ we have:

a) $a_n \in\mathbb{N}$

b) $a_n\lt a_{n+1}$

c) Exists $\displaystyle\lim_{n\to \infty} \frac{\#\{j\mid a_j\le n\}}{n}$

Let $A$ the set of the sequences that meet the aforementioned conditions. Which is the cardinal of $A$?

I have no idea, I don't known how to interpret the last condition. Any hint?

Answer 1

It's uncountable. Take any real number between 0 and 1 and make a sequence of your type out of the decimal as in:

$$\alpha = .912092013.... \;\;\;a_1 = 9 \;a_2=91\;\;a_3=912\;\;a_4 =9120$$ And so on. Clearly increasing and there is only one $a_i$ between any two consecutive powers of 10.

(in case of ambiguity for terminating rationals, use the expansion that ends with a string of $0$'s)