Number of points of accumulation of a sequence

Can a sequence have infinitely many points of accumulation i.e. we can extract infinitely many subsequences from it s.t. they all converge to their respective point of accumulation? I have the feeling it would mean that the period of repetition of something could be infinitely big.

Solution 1:

Start with $0,1$. Then travel backwards to $0$ in steps of $1/2$, so $1/2,0$. Then travel forwards to $1$ in steps of $1/4$, so $1/4,2/4,3/4,1$. Then travel backwards to $0$ in steps of $1/8$, so $7/8$, $6/8$, $5/8$, and so on. Continue.

Every real between $0$ and $1$ is an accumulation point of our sequence.

Solution 2:

Yes, this is possible. For example consider the sequence $a_n$ for $n \ge 2$ defined as the smallest divisor of $n$ greater than $1$.

The accumulation points are all the prime numbers. Subsequences witnessing them are for instance the $p$-th powers for each prime $p$.

Solution 3:

The rationals are a countable set. We can define a 1-to-1 function $f:N\to Q$ with $Q=\{f(n):n\in N\}.$ Consider the sequence $S=(f(n))_{n\in N}.$ Every real number is a limit point of a subsequence of $S.$