How to define a sequence with an infinite amount of each natural number? I was trying to create a sequence such that for every $x \in \mathbb{N}$, that sequence has a subsequence that converges to $x$. 
Basically I came up with the sequence that has each natural number an infinite amount of times. Now I am having trouble defining it. Any help?

One general way to create a sequence with an infinite number of accumulation points
 is to write a table of your sequence and read it by anti-diagonals. An intuitive way of looking at your problem is to write down the sequence in this form:
$$
1,\\
1,2,\\
1,2,3,\\
1,2,3,4,\\
\vdots \vdots
$$
