I'm little bit confused about limit arithmetic:
$$\lim _{n\to \infty }\left(1\right)\:=\:\lim _{n\to \infty }\left(n\cdot \frac{1}{n}\right)\:=\:\lim _{n\to \infty }\left(\frac{1}{n}\:+\:\frac{1}{n}\:+\:...+\frac{1}{n}\right)\:=\\\lim _{n\to \infty }\left(\frac{1}{n}\right)+\lim _{n\to \infty }\left(\frac{1}{n}\right)+....+\lim _{n\to \infty }\left(\frac{1}{n}\right)\:=\\0\:+0\:+...\:+0\:=\:0$$
what i'm missing in the arithmetic?
The equality
$$\lim_{n\to\infty} \left(\frac1n+\frac1n +\cdots + \frac1n\right) = \lim_{n\to\infty}\frac 1n+\lim_{n\to\infty}\frac 1n+\cdots +\lim_{n\to\infty}\frac 1n$$
is not true.
What is true is this:
If you have $k$ sequences of real numbers, $\left(a_n^{(1)}\right)_{n\in\mathbb N}, \left(a_n^{(2)}\right)_{n\in\mathbb N}, \dots, \left(a_n^{(k)}\right)_{n\in\mathbb N}$ and all these sequences converge, then
$$\lim_{n\to\infty} (a_n^{(1)} + a_n^{(2)}+\cdots + a_n^{(k)}) = \lim_{n\to\infty}a_n^{(1)} + \lim_{n\to\infty}a_n^{(2)} + \cdots \lim_{n\to\infty}a_n^{(k)}$$
but that is not what you have. What you have is an infinite amount of sequences. For each $k\in\mathbb N$, you have a sequence $\left(a_n^{(k)}\right)_{n\in\mathbb N}$, where $a_n^{(k)}$ is defined as
$$ a_n^{(k)} = \begin{cases} 0 & \mbox{ if } k>n\\ \frac1n&\mbox{ if } k\leq n \end{cases}. $$ Then, you want to say that
$$\lim_{n\to\infty}\left(a_n^{(1)} + a_n^{(2)} + \cdots \right) = \lim_{n\to\infty}a_n^{(1)} + \lim_{n\to\infty}a_n^{(2)} + \lim_{n\to\infty}a_n^{(3)} + \cdots$$ which is not true in all cases, as your example clearly shows.