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.