A Question on Cardinality $\aleph_{0}$ I'm trying to understand the concept of Cardinality.
My question is, 
Let the interval $[1, 2n]$ is given.
In this interval we have $2n$ natural numbers. Or $n\to\infty$, we have countable infinite natural numbers and Cardinality equal to $\aleph_0$.
Then, in this interval we have $n$ even natural numbers. Or $n\to\infty$, we have countable infinite even natural numbers and Cardinality equal to $\aleph_0$.
Then for $n\to\infty$, in this interval $[1,2n]$,we have $$\lim_{n\to\infty} \frac {\text{number of even natural numbers}}{\text{number of all natural numbers}}=\frac 12.$$ In other words number of natural numbers $2$ times many from number of even natural numbers. But, why the cardinalities are equal or What's the point I confused?

Perhaps $\infty=\frac12\cdot\infty$ seems wrong, but $0=\frac12\cdot0$ should illustrate that there are numbers that remain unchanged when divided by $2$. It is possible that $2\cdot\infty=\infty$. What you describe above means that the "density" of the set of even natural numbers is $\frac12$, but the concept of density is different from that of cardinality. https://en.wikipedia.org/wiki/Natural_density One uses limits to define density, but only a bijection (no reference to finite subsets) to define equal cardinality of two infinite sets. The function $f(n)=2n$ provides such a bijection from the set of all natural numbers $\ge1$ to the set of all even natural numbers $\ge2$, so these two sets have the same cardinality.   
