To show that if $(a_n)_{n \in \mathbb{N}}$ and $(b_n)_{n \in \mathbb{N}}$ are equivalent sequences, then $(a_n)_{n \in \mathbb{N}}$ converges if and only if $(b_n)_{n \in \mathbb{N}}$ converges, we need to first understand the definition of equivalent sequences.

Definition of Equivalent Sequences:

Two sequences $(a_n)$ and $(b_n)$ are said to be equivalent if:

$\lim_{n \to \infty} (a_n - b_n) = 0$

This means that as $n$ becomes arbitrarily large, the difference between $a_n$ and $b_n$ becomes arbitrarily small.

Proof:

We will show both directions of the statement.

1. If $(a_n)$ converges, then $(b_n)$ converges:

Let $(a_n)$ converge to some limit $L \in \mathbb{R}$, i.e.,

$\lim_{n \to \infty} a_n = L.$

We are also given that $(a_n)$ and $(b_n)$ are equivalent, which implies:

$\lim_{n \to \infty} (a_n - b_n) = 0.$

Thus, we can rewrite $b_n$ as:

$b_n = a_n - (a_n - b_n).$

Now, since $\lim_{n \to \infty} a_n = L$ and $\lim_{n \to \infty} (a_n - b_n) = 0$, by the sum of limits property, we have:

$\lim_{n \to \infty} b_n = \lim_{n \to \infty} a_n - \lim_{n \to \infty} (a_n - b_n) = L - 0 = L.$

Thus, $(b_n)$ converges to $L$.

2. If $(b_n)$ converges, then $(a_n)$ converges:

Let $(b_n)$ converge to some limit $L \in \mathbb{R}$, i.e.,

$\lim_{n \to \infty} b_n = L.$

Since $(a_n)$ and $(b_n)$ are equivalent, we again know:

$\lim_{n \to \infty} (a_n - b_n) = 0.$

Now, we can rewrite $a_n$ as:

$a_n = b_n + (a_n - b_n).$

Using the sum of limits property again, we have:

$\lim_{n \to \infty} a_n = \lim_{n \to \infty} b_n + \lim_{n \to \infty} (a_n - b_n) = L + 0 = L.$

Thus, $(a_n)$ converges to $L$.

Conclusion:

Since we've shown both directions, we conclude that:

$(a_n)_{n \in \mathbb{N}} \text{ converges if and only if } (b_n)_{n \in \mathbb{N}} \text{ converges}.$

Would you like more details or have any questions?

Here are 5 related questions:

1. How do we define the convergence of a sequence rigorously?
2. Can you give examples of two equivalent sequences?
3. What is the sum of limits property used in the proof?
4. How can we visualize the concept of equivalent sequences?
5. Does the proof hold if the sequences are complex-valued instead of real-valued?

Tip: Equivalent sequences must get arbitrarily close as $n$ grows large, but they don't need to be identical at every step.