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?

First of all, you try to work with limits and want to use that an expression at the limit point equals the limit of said expression as we approach the limit point. But for that you first of all need to know that the function you consider is defined at the limit point. So, how do you define division at infinity? And even if defined, you'd need continuity for your suggested conclusion. E.g., exponentiation $(x,y)\mapsto x^y$ is defined at $(0,0)$, namely $0^0=1$. However, exponentiation is not continous there and therefore we cannot infer $\lim x_n^{y_n}=1$ from $\lim x_n=\lim y_n=0$.
That being said, you should not blindly assume that well-known properties of arithmetic for finite numbers transfer readily to the arithmetic of infinite cardinalities or ordinals.
Final remark: Did you notice that you want to make a claim about $\color{red}{\aleph_0}$ but that limits use the notation $\lim_{x\to{\color{red}\infty}}$ instead?
