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In mathematics, a sequence is an ordered list of elements that typically follow a certain rule or pattern. Each element in the sequence is called a term, and we can refer to them by their position in the list. For example, the sequence of even natural numbers could be represented as 2, 4, 6, 8,... where each term is the previous term plus 2.

A sequence can be finite or infinite. A finite sequence has a specific number of terms, while an infinite sequence continues indefinitely. For instance, the set of all multiples of a number is usually an infinite sequence. Sequences can be given by a closed-form formula, where each term can be computed from its index, or defined by a recurrence relation where each term is derived from the preceding terms.

Sequences are fundamental in various branches of mathematics where they are used to model real-life situations and solve mathematical problems. From the perspective of analysis, they also become a stepping stone to the concept of limits and convergence.

In the world of sequences, a subsequence takes on the role of a 'selected highlight' from an original sequence. This subsequence is formed by taking elements from the original sequence without changing their order, but not necessarily taking every element. For example, from the sequence 1, 3, 5, 7,... an example subsequence would be 1, 5, 7,... Here, we've skipped over the term 3, but maintained the original order of the selected terms.

A subsequence is considered infinite if it is derived from an infinite sequence and contains infinitely many terms. Importantly, while every sequence is a potential subsequence of itself, it's the subsequences that do not include all the terms of the original sequence that often get our attention, especially when exploring properties like convergence.

Diving into the heart of a convergent subsequence, this is essentially a subsequence that has the extraordinary property of approaching a specific value as the number of terms grows. The limit of a subsequence is the value that the terms get closer to, and when the distance between the terms and this value can be made smaller than any pre-defined positive number, we deem the subsequence convergent.

For example, consider the subsequence 1/2, 1/4, 1/8, 1/16,... Each term is half the previous term, which means the terms are getting closer and closer to zero. We say this subsequence is convergent with the limit being 0. What is fascinating here is that even within sequences that diverge or are unbounded, we can sometimes find these pockets of order – convergent subsequences that march steadily towards a finite limit.