This question delves into the fundamental properties of metric spaces, specifically the relationship between Cauchy sequences and convergent sequences.
Step 1: Understanding the Question:
We need to determine if the property of being a Cauchy sequence (terms getting closer to each other) is sufficient to guarantee that the sequence converges to a limit *within the given metric space*.
Step 2: Key Formula or Approach:
This question is about the definition of a complete metric space. A metric space is defined as complete if and only if every Cauchy sequence in it converges to a limit within that space.
Step 3: Detailed Explanation:
A sequence \(\{x_n\}\) is Cauchy if for any distance \(\epsilon>0\), there exists an integer \(N\) such that for all \(m, n>N\), the distance \(d(x_m, x_n)<\epsilon\).
A sequence \(\{x_n\}\) is convergent if there exists a point \(L\) in the space such that for any \(\epsilon>0\), there exists an \(N\) such that for all \(n>N\), \(d(x_n, L)<\epsilon\).
In any metric space, every convergent sequence is a Cauchy sequence. However, the converse is not always true.
Counterexample: Consider the metric space of rational numbers \( \mathbb{Q} \) with the usual distance \(d(x,y) = |x-y|\). The sequence of rational numbers \(x_1=3, x_2=3.1, x_3=3.14, \dots\) that approximates \( \pi \) is a Cauchy sequence. However, it does not converge *in \( \mathbb{Q} \)* because its limit, \( \pi \), is an irrational number and is not an element of the space \( \mathbb{Q} \).
Therefore, a Cauchy sequence is not necessarily convergent in a general metric space. This property only holds in complete metric spaces like \( \mathbb{R} \) or \( \mathbb{C} \).
Step 4: Final Answer:
No, a Cauchy sequence is not always convergent in an arbitrary metric space.