Why sequence 1/n fail to converge in the set of all positive real numbers with the metric space d(x,y)=|x-y|

The sequence \( \frac{1}{n} \) does not converge in the set of all positive real numbers \( \mathbb{R}^+ \) with the metric space \( d(x, y) = |x - y| \) because its limit lies outside the set \( \mathbb{R}^+ \).

### Explanation:
1. **Definition of Convergence**: A sequence \( \{x_n\} \) in a metric space \( (X, d) \) converges to a point \( x \in X \) if for every \( \epsilon > 0 \), there exists an integer \( N \) such that for all \( n \geq N \), \( d(x_n, x) < \epsilon \). In simpler terms, the sequence \( x_n \) gets arbitrarily close to \( x \) as \( n \) increases.

2. **Behavior of the Sequence \( \frac{1}{n} \)**: The sequence \( \left\{\frac{1}{n}\right\} \) where \( n \) is a positive integer, is a decreasing sequence that approaches 0 as \( n \) tends to infinity. Mathematically, \( \lim_{n \to \infty} \frac{1}{n} = 0 \).

3. **Set of Positive Real Numbers \( \mathbb{R}^+ \)**: The set \( \mathbb{R}^+ \) consists of all real numbers greater than 0. Notably, 0 is not included in this set.

4. **Convergence Issue**: The sequence \( \left\{\frac{1}{n}\right\} \) converges to 0, which is not an element of \( \mathbb{R}^+ \). Since the limit point 0 does not belong to the set \( \mathbb{R}^+ \), the sequence \( \left\{\frac{1}{n}\right\} \) cannot converge within \( \mathbb{R}^+ \).

Thus, the sequence \( \frac{1}{n} \) fails to converge in \( \mathbb{R}^+ \) because its limit lies outside the space.

Do you want further details or have any other questions?

Here are some related questions:
1. What happens to a sequence if its limit lies outside the metric space?
2. Can a sequence be Cauchy but not converge in \( \mathbb{R}^+ \)?
3. How does the concept of a limit differ in closed vs. open sets?
4. What is the difference between convergence and divergence in a metric space?
5. How does the completeness of a space relate to convergence?

**Tip:** In a metric space, a sequence can only converge to a point within the space. If its limit lies outside the space, the sequence does not converge in that space.

Understand why the sequence 1/n fails to converge in the set of all positive real numbers with the metric space d(x, y) = |x - y|. Explore the behavior of the sequence, its limit, and the properties of the set of positive real numbers. Discover the implications of convergence issues outside the metric space.

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