To prove that every convergent sequence in a metric space is a Cauchy sequence, let's go through it step by step.

Definitions:

1. Convergent Sequence: A sequence $(x_n)$ in a metric space $(X, d)$ converges to a point $x \in X$ if for every $\epsilon > 0$, there exists a natural number $N$ such that for all $n \geq N$, $d(x_n, x) < \epsilon$.

2. Cauchy Sequence: A sequence $(x_n)$ in a metric space $(X, d)$ is called a Cauchy sequence if for every $\epsilon > 0$, there exists a natural number $N$ such that for all $m, n \geq N$, $d(x_n, x_m) < \epsilon$.

Proof:

Let $(x_n)$ be a sequence that converges to $x \in X$ in the metric space $(X, d)$. We need to show that $(x_n)$ is a Cauchy sequence.

Step 1: Convergence implies small distances to the limit

Since $(x_n)$ converges to $x$, by definition: $\forall \epsilon > 0, \exists N \in \mathbb{N} \text{ such that } \forall n \geq N, \ d(x_n, x) < \frac{\epsilon}{2}.$ That is, for all $n \geq N$, the distance between $x_n$ and $x$ is less than $\frac{\epsilon}{2}$.

Step 2: Use the triangle inequality

Now, consider any $m, n \geq N$. We want to show that $d(x_n, x_m) < \epsilon$. Using the triangle inequality: $d(x_n, x_m) \leq d(x_n, x) + d(x_m, x).$ Since $n, m \geq N$, we know from Step 1 that both $d(x_n, x) < \frac{\epsilon}{2}$ and $d(x_m, x) < \frac{\epsilon}{2}$. Substituting these into the inequality: $d(x_n, x_m) \leq \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon.$

Step 3: Conclusion

Therefore, for all $m, n \geq N$, we have $d(x_n, x_m) < \epsilon$, which is precisely the definition of a Cauchy sequence.

Thus, we have shown that every convergent sequence in a metric space is a Cauchy sequence.

Would you like further details or have any questions?

Relative questions:

1. Can you give an example of a sequence that is Cauchy but not convergent?
2. What is the difference between Cauchy and convergent sequences in non-complete metric spaces?
3. How does the proof change if the metric space is not complete?
4. Can you explain the role of the triangle inequality in this proof?
5. What are some examples of metric spaces where every Cauchy sequence converges?

Tip:

In any metric space, a Cauchy sequence doesn’t always have to converge unless the space is complete. However, every convergent sequence is always Cauchy.