TITLE: Percentage of natural numbers that are perfect squares?
QUESTION [23 upvotes]: While reading about the prime number distribution I came across this fact that the percentage of natural numbers that are perfect square is zero. How do I prove this ?

REPLY [30 votes]: Zero percent of all natural numbers are perfect squares in the sense that the limit of the proportion of the perfect squares  to natural numbers is zero.To express it, consider
$$\lim_{x\rightarrow\infty} \frac{\left|\left\{n\in\mathbb{N}\mid n≤x \wedge n\ \mbox{is a perfect square}\right\}\right|}x.$$
Since the numerator is $\lfloor\sqrt{x}\rfloor$, thus approximately $\sqrt x$ and
$$\lim_{x\rightarrow\infty} \frac{\sqrt{x}}x = \lim_{x\rightarrow\infty} \frac{1}{\sqrt{x}} = 0,$$
the statement follows that zero percent of all natural numbers are perfect squares.