Consider the following sequences, and determine if they converge or not in Q.
1. (2)n∈N
2. 
2 − 1
n

n∈N

Let's analyze both sequences to determine if they converge in **Q** (the set of rational numbers).

### 1. Sequence \( (2)_n \in \mathbb{N} \)

This is a constant sequence where every term is 2. So, for all \( n \in \mathbb{N} \),
\[
a_n = 2
\]
Since the sequence is constant, it converges to the value 2. The limit is 2, which is a rational number. Therefore, this sequence **converges** in \( \mathbb{Q} \).

### 2. Sequence \( \left( 2 - \frac{1}{n} \right)_n \in \mathbb{N} \)

For this sequence, each term is given by:
\[
a_n = 2 - \frac{1}{n}
\]
As \( n \to \infty \), \( \frac{1}{n} \to 0 \), so:
\[
a_n \to 2 - 0 = 2
\]
Thus, the sequence converges to 2. Since 2 is a rational number, this sequence also **converges** in \( \mathbb{Q} \).

### Conclusion:
Both sequences converge to 2, and since 2 is a rational number, both sequences converge in \( \mathbb{Q} \).

Would you like further details or explanations?

Here are 5 related questions:
1. What conditions are necessary for a sequence to converge in \( \mathbb{Q} \)?
2. Can a sequence of irrational numbers converge to a rational number?
3. How would the analysis change if we considered these sequences in \( \mathbb{R} \)?
4. What is the difference between a constant sequence and a non-constant converging sequence?
5. How does the definition of convergence change in different number systems (e.g., \( \mathbb{C} \), \( \mathbb{R} \))?

**Tip:** When analyzing sequence convergence, always check the behavior as \( n \to \infty \) and consider whether the limit is within the given set.

Consider the following sequences, and determine if they converge or not in Q. 1. (2)n∈N 2. (2 − 1/n)n∈N

Analyze the convergence of two sequences in the set of rational numbers (Q): a constant sequence and a sequence approaching a limit. Learn about sequence convergence, rational limits, and key mathematical concepts like limits and convergence definitions. Suitable for undergraduate mathematics students.

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