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.