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?

How about just:
$$ a_n=1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,\ldots$$
Clearly each natural number appears an infinite amount of times, so you can extract a subsequence converging to whichever natural number you want.
