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How do I find asymptotic density of the set of even numbers $\{2,4,6,\cdots\}$?

I know that for a subset of the natural numbers $A⊆ℕ$ let $a(n)$ be the number of elements of A which are less than or equal to n.

i.e. $a(n)=\|A∩\{1,2,⋯,n\}\|$.

Then, the asymptotic density of a subset of $\mathbb N$ is defined as $\lim_{n→∞}a(n)/n$.

But I'm unsure how to calculate still. Thanks.

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  • $\begingroup$ It is $1/2$. Evaluating it is not too hard. (Divide the case of $n$!) $\endgroup$ – Hanul Jeon May 21 '17 at 9:53
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The right answer is $1/2$, and I hope this matches your intuition. In fact you can calculate that $a(n) = n/2$ if $n$ is even and $a(n) = (n-1)/2$ if $n$ is odd. This implies that for all natural $n$ we have: $$\frac{1}{2}-\frac{1}{2n} \leq \frac{a(n)}{n} \leq \frac{1}{2}$$ So the limit is indeed $1/2$ as $n\rightarrow \infty$.

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