To demonstrate that the sequence $a_n = \frac{1}{n}$ is convergent, we need to show that it approaches a particular limit as $n \to \infty$.

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Step 1: Definition of Convergence

A sequence $a_n$ is said to converge to a limit $L$ if, for every $\epsilon > 0$, there exists a positive integer $N$ such that for all $n \geq N$, $|a_n - L| < \epsilon.$

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Step 2: Sequence Analysis

The given sequence is: $a_n = \frac{1}{n}.$ We claim that $a_n$ converges to $L = 0$.

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Step 3: Proof

Let $\epsilon > 0$ be arbitrary. We need to find $N \in \mathbb{N}$ such that: $\left| \frac{1}{n} - 0 \right| < \epsilon \quad \text{for all } n \geq N.$ Simplifying this inequality: $\frac{1}{n} < \epsilon.$ By rearranging, this implies: $n > \frac{1}{\epsilon}.$

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Step 4: Choosing $N$

Let $N = \lceil \frac{1}{\epsilon} \rceil$, where $\lceil x \rceil$ denotes the ceiling function (the smallest integer greater than or equal to $x$). Then, for all $n \geq N$, we have: $n \geq \lceil \frac{1}{\epsilon} \rceil \geq \frac{1}{\epsilon}.$ This ensures: $\frac{1}{n} \leq \frac{1}{\lceil \frac{1}{\epsilon} \rceil} \leq \epsilon.$

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Step 5: Conclusion

By the definition of convergence, $\frac{1}{n} \to 0$ as $n \to \infty$.

Hence, the sequence $\frac{1}{n}$ is convergent, and its limit is $0$.

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Would you like more details on this proof or explore related concepts?

Here are some questions to extend your understanding:

1. How does this proof differ for a general series instead of a sequence?
2. Can you apply this reasoning to other sequences, such as $\frac{1}{n^2}$ or $\frac{1}{\sqrt{n}}$?
3. What is the formal difference between convergence of a sequence and convergence of a function?
4. How does this sequence behave when plotted on a graph as $n \to \infty$?
5. How would divergence of a sequence be proven for a sequence like $n$?

Tip: When proving convergence, always state and verify the limit using the formal definition of a limit.