Existence of a sequence that has every element of $\mathbb N$ infinite number of times

Solution 1:

$$\langle \underbrace{0}_{1\text{ term}},\underbrace{0,1}_{2\text{ terms}},\underbrace{0,1,2}_{3\text{ terms}},\underbrace{0,1,2,3}_{4\text{ terms}},\underbrace{0,1,2,3,4}_{5\text{ terms}},\dots\rangle$$

Each $n\in\Bbb N$ appears in all but the first $n$ blocks, hence infinitely often.

Solution 2:

Let $a_n$ be the largest natural number, $k$ such that $2^k$ divides $n+1$

Solution 3:

Let $f\colon\Bbb{N\to N\times N}$ be a bijection, and let $a_n$ be the right coordinate of $f(n)$, then $\langle a_n\mid n\in\Bbb N\rangle$ is a sequence covering all the elements of $\Bbb N$, and each appearing infinitely many times.