The sequence un=1−1n
, for n integer >0,
 has a limit when n →+∞
 because

Select one:

it is decreasing and it has a lower bound

None of the proposed choices


it is increasing and it has a lower bound

it is increasing and it has an upper bound

it is decreasing and it has an upper bound

To analyze the sequence \( u_n = 1 - \frac{1}{n} \) for \( n > 0 \):

1. **Limit Calculation**: As \( n \to +\infty \), \( \frac{1}{n} \to 0 \), so \( u_n \to 1 \).

2. **Behavior of the Sequence**:
   - **Increasing or Decreasing**: 
     - For \( n \) and \( n+1 \), we compare:
       \[
       u_{n+1} = 1 - \frac{1}{n+1}
       \]
       \[
       u_n = 1 - \frac{1}{n}
       \]
       To see if \( u_{n+1} > u_n \):
       \[
       1 - \frac{1}{n+1} > 1 - \frac{1}{n} \quad \Rightarrow \quad -\frac{1}{n+1} > -\frac{1}{n} \quad \Rightarrow \quad \frac{1}{n+1} < \frac{1}{n}
       \]
       This inequality holds, indicating that \( u_n \) is increasing.

3. **Boundedness**:
   - **Lower Bound**: \( u_n \) is bounded below by 0 since \( u_n = 1 - \frac{1}{n} > 0 \) for \( n \geq 1 \).
   - **Upper Bound**: \( u_n < 1 \) for all \( n > 0 \).

### Conclusion
The correct choice is:
**it is increasing and it has an upper bound.**

Would you like more details on this topic or have any questions? Here are some related questions for further exploration:

1. How do you determine if a sequence is bounded?
2. What is the formal definition of a limit?
3. Can a sequence be both increasing and decreasing?
4. What other sequences converge to 1?
5. How can the monotonicity of a sequence affect its limit? 

**Tip**: Always check the definition of increasing and decreasing sequences to analyze their behavior effectively.

The sequence un=1−1/n, for n integer >0, has a limit when n →+∞ because

Explore the behavior of the sequence un=1−1/n as n approaches infinity. This page covers the concepts of increasing sequences, boundedness, and the limit of the sequence, suitable for high school mathematics students.

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