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The sequence \( U_n \) is defined as follows:

\[
U_n = 
\begin{cases}
n & \text{se } n < 10 \\
1 + \frac{1}{n} & \text{se } n \geq 10
\end{cases}
\]

### Step 1: Analyze for \( n < 10 \)
For \( n < 10 \), the sequence is simply \( U_n = n \). This is strictly increasing because as \( n \) increases, \( U_n \) also increases. The values are unbounded below 10.

### Step 2: Analyze for \( n \geq 10 \)
For \( n \geq 10 \), the sequence becomes \( U_n = 1 + \frac{1}{n} \). Since \( \frac{1}{n} \) decreases as \( n \) increases, \( U_n \) will approach 1 as \( n \) approaches infinity. Therefore, for \( n \geq 10 \), the sequence is strictly decreasing and converges to 1.

### Step 3: Monotonicity
- **For \( n < 10 \)**: The sequence \( U_n \) is strictly increasing.
- **For \( n \geq 10 \)**: The sequence \( U_n \) is strictly decreasing.

Thus, the sequence is not monotonic overall because it increases for \( n < 10 \) and decreases for \( n \geq 10 \).

### Step 4: Boundedness
- For \( n < 10 \), \( U_n = n \) takes values from 1 to 9, which are bounded.
- For \( n \geq 10 \), \( U_n = 1 + \frac{1}{n} \), which is always greater than 1 but tends to 1 as \( n \to \infty \).

Thus, the sequence is **bounded** for all \( n \).

### Step 5: Convergence or Divergence
- For \( n < 10 \), \( U_n \) does not converge since it is simply increasing.
- For \( n \geq 10 \), \( U_n \) converges to 1 as \( n \to \infty \).

Therefore, the sequence is **bounded** and **partially convergent** (for \( n \geq 10 \), it converges to 1), but it is not monotonic. It is **not divergent** as the sequence has a bounded behavior.

### Final Answer:
The sequence \( U_n \) is **bounded** and **partially convergent**, but it is not monotonic. It does not diverge.

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Do you have any questions or need further clarification?

Here are 5 related questions:
1. What does it mean for a sequence to be monotonic?
2. How do we prove that a sequence converges?
3. What are the conditions for a sequence to be bounded?
4. What is the difference between divergent and convergent sequences?
5. How can we analyze the limit of a sequence defined piecewise like this one?

**Tip:** When analyzing piecewise sequences, always study the behavior in each defined interval separately before combining the results!

Seja a sucessão definida por U_n = n, se n < 10, e U_n = 1 + 1/n, se n ≥ 10. A sucessão U_n é limitada ou divergente ou monótona? Justifique a sua resposta.

Learn how to analyze the sequence U_n defined piecewise with U_n = n for n < 10 and U_n = 1 + 1/n for n ≥ 10. Understand the concepts of monotonicity, boundedness, and convergence with a detailed step-by-step solution. Suitable for high school students studying advanced algebra.

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