Determine all Convergent Subsequences (with their limit) of the Sequence $1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, \dots$ [closed]

Problem: Determine all convergent subsequences (with their limit) of the sequence $1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, \dots$

I know that a subsequence of $\{a\}$ is a sequence $\{b\}$ defined by $b_k=a_{n_k}$, where $n_1<n_2<\dots$ is an increasing sequence of indices.As an example of a subsequence of {a}, we can consider the constant sequences equal to $1, 2,$ or $3$; all these sequences are convergent. However, I don't know how to determine all possible subsequences. I can't figure out how to proceed to identify them.

Could someone help me understand how to find all the subsequences of a given sequence?