To determine the set $L$ of all limit points of the sequence $(z_n)_{n \in \mathbb{N}}$, we analyze its definition and the given conditions.

Step 1: Understanding the Sequence Definition

We define the sequence $(z_n)_{n \in \mathbb{N}}$ using the sequences $(a_{n,r})_{n \geq 0}$ for $r = 1, \dots, q$. Specifically, the value of $z_n$ depends on how $n$ is expressed in terms of $k$ and $r$:

$z_n = a_{k,r} \quad \text{if} \quad n = kq + r, \quad \text{where} \quad k \in \mathbb{N} \cup \{0\}, \quad 1 \leq r \leq q.$

This means that the sequence $(z_n)$ is formed by interleaving the subsequences $(a_{n,r})$, each converging to $b_r$.

Step 2: Identifying the Limit Points

A limit point of $(z_n)$ is any real number that is the limit of some subsequence of $(z_n)$.

• Each sequence $(a_{n,r})$ is given to be convergent with:

  $a_{n,r} \to b_r \text{ as } n \to \infty.$

• Since the sequence $(z_n)$ is composed of values from all $(a_{n,r})$, we examine its subsequences corresponding to each $r$.

• Fixing $r$, the subsequence $(a_{k,r})_{k \geq 0}$ (where $k$ indexes every $q$-th term of $z_n$) converges to $b_r$.

• Therefore, for each $r = 1, \dots, q$, $b_r$ is a limit point of $(z_n)$.

• Moreover, since $(z_n)$ is entirely composed of elements from $(a_{n,r})$, the only possible limits are the limits of these sequences.

Step 3: Conclusion

Since the only possible limit points are $b_1, b_2, \dots, b_q$, we conclude that:

$L = \{b_1, b_2, \dots, b_q\}.$

Thus, the set of all limit points of $(z_n)$ is precisely the set of limit values of the original sequences $(a_{n,r})$.

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Would you like a more detailed breakdown of any part? Here are five related questions to deepen understanding:

1. Why do the subsequences $(a_{k,r})$ preserve the convergence properties of $(a_{n,r})$?
2. Can $(z_n)$ have a limit different from any $b_r$?
3. How does the choice of $q$ influence the structure of $(z_n)$?
4. What would change if not all sequences $(a_{n,r})$ converged?
5. How can this result be generalized for more complex sequences?

Tip: When analyzing sequences, always consider subsequences corresponding to modular arithmetic patterns, as they often reveal convergence behavior.