Question 1: Suppose (an) is a sequence defined by an =
1
n+1 for n = 1, 2, 3, . . ..
1. Prove that the sequence (an) converges.
2. Find the limit of the sequence as n → ∞.
3. Explain intuitively why this limit makes sense in terms of the behavior of the sequence’s terms as
n increases.

To address this problem comprehensively, let's tackle each part step by step.

### Part 1: Prove that the sequence \((a_n)\) converges

The sequence \((a_n)\) is given by:
\[
a_n = \frac{1}{n+1}
\]
for \( n = 1, 2, 3, \ldots \).

To prove that \((a_n)\) converges, let's analyze its behavior as \( n \) becomes very large.

1. **Monotonicity**: Notice that \( a_n = \frac{1}{n+1} \) is positive for all \( n \geq 1 \) and decreases as \( n \) increases. To confirm this, observe that \( a_{n+1} = \frac{1}{n+2} \), and since \( n+2 > n+1 \), it follows that \( \frac{1}{n+2} < \frac{1}{n+1} \). Therefore, \((a_n)\) is a decreasing sequence.

2. **Boundedness**: Since \( a_n = \frac{1}{n+1} \) is always positive, we know \( a_n \geq 0 \) for all \( n \geq 1 \). Therefore, \((a_n)\) is bounded below by \( 0 \).

3. **Conclusion**: By the Monotone Convergence Theorem, any sequence that is both decreasing and bounded below must converge. Therefore, \((a_n)\) converges.

### Part 2: Find the limit of the sequence as \( n \to \infty \)

To find the limit, we examine \( \lim_{n \to \infty} a_n \):
\[
\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{1}{n+1}.
\]

As \( n \to \infty \), the term \( n+1 \) grows indefinitely, causing \( \frac{1}{n+1} \) to approach \( 0 \). Thus, we conclude:
\[
\lim_{n \to \infty} a_n = 0.
\]

### Part 3: Intuitive Explanation of the Limit

Intuitively, as \( n \) increases, the denominator \( n+1 \) in the expression \( \frac{1}{n+1} \) becomes larger and larger, making the fraction smaller. This trend indicates that each term in the sequence becomes closer to zero as \( n \) grows. The limit of zero aligns with this intuition, as the terms of the sequence approach zero in value as they progress, illustrating a common pattern in sequences where the reciprocal of a linearly increasing term leads to a zero limit.

In summary:
1. The sequence \((a_n)\) converges because it is decreasing and bounded below.
2. The limit of \((a_n)\) as \( n \to \infty \) is \( 0 \).
3. Intuitively, this makes sense because the terms get smaller as \( n \) increases due to the growing denominator.

Would you like further clarification on any of these steps?

Here are five related questions that might help deepen your understanding:

1. Can you prove convergence for other types of sequences, such as \((b_n) = \frac{1}{n^2+1}\)?
2. How does the limit of a sequence differ from a limit of a function as \( x \to \infty \)?
3. What happens if you alter the sequence definition to \((c_n) = \frac{2}{n+1}\)?
4. Why is the Monotone Convergence Theorem helpful for proving convergence?
5. How do we approach convergence proofs for non-monotonic sequences?

**Tip:** In many cases, you can determine a sequence's limit by examining how its formula behaves as \( n \) approaches infinity.

Question 1: Suppose (an) is a sequence defined by an = 1 / (n + 1) for n = 1, 2, 3, .... 1. Prove that the sequence (an) converges. 2. Find the limit of the sequence as n → ∞. 3. Explain intuitively why this limit makes sense in terms of the behavior of the sequence’s terms as n increases.

Explore the convergence of the sequence an = 1 / (n + 1) with a detailed proof. This solution includes an intuitive explanation of why the sequence approaches zero, key formulas, and the Monotone Convergence Theorem, making it ideal for undergraduate-level understanding.

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