The problem requires you to prove that the sequence $(a_n)$ with $a_n = \sqrt{n} - \sqrt{n-1}$ for all $n \in \mathbb{N}$ is convergent and to determine its limit.

Proof of Convergence

1. Expression Simplification: To analyze $a_n$, we rationalize the expression: $a_n = \sqrt{n} - \sqrt{n-1}$ Multiply and divide by the conjugate $\sqrt{n} + \sqrt{n-1}$:

   = \frac{n - (n-1)}{\sqrt{n} + \sqrt{n-1}}$$ Simplify the numerator: $$a_n = \frac{1}{\sqrt{n} + \sqrt{n-1}}$$

2. Limit Analysis: As $n \to \infty$, the denominator $\sqrt{n} + \sqrt{n-1}$ grows without bound because $\sqrt{n} \to \infty$. Therefore, the fraction tends to $0$. Formally: $\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{1}{\sqrt{n} + \sqrt{n-1}} = 0$

3. Convergence: Since $a_n \to 0$ as $n \to \infty$, the sequence $(a_n)$ is convergent, with the limit: $\lim_{n \to \infty} a_n = 0$

Conclusion

The sequence $(a_n)$ is convergent, and its limit is: $\boxed{0}$

Let me know if you'd like more details or further explanations.

Here are five related questions to expand your understanding:

1. How does rationalizing a sequence help in determining its limit?
2. What is the role of asymptotic behavior in determining convergence?
3. Can you apply the same method to analyze the sequence $a_n = \sqrt{n+1} - \sqrt{n}$?
4. What are other methods for proving convergence of sequences involving square roots?
5. How can we formally prove that a sequence is bounded and monotonic to show convergence?

Tip: Always simplify sequences involving square roots by rationalizing them—it often reveals the asymptotic behavior of the terms!