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A subsequence is derived from an original sequence by picking some of its elements, which are then arranged in the same order as they appear in the original sequence. Consider a sequence \( \langle a_n \rangle \), from which you select specific terms using a function \( f(n) \). This function maps each natural number \( n \) to a term index in the sequence and is strictly increasing, ensuring that as \( n \) grows, \( f(n) \) also grows while maintaining the order of the elements. Essentially, a subsequence keeps the original sequence's order but may skip some terms.

Understanding subsequences helps when analyzing the behavior of a sequence as we can focus on particular elements that retain the properties of the whole due to their ordered selection. A vital property of subsequences is that if the entire sequence is convergent, the subsequence will have the same limit.

The concept of a limit is central to understanding convergence in sequences. For a sequence \( \langle a_n \rangle \) to be convergent, it should approach a certain value known as the limit. Mathematically, a sequence has a limit \( L \) if for every \( \epsilon > 0 \), however small, there is a natural number \( N \) such that for all terms where \( n \geq N \), the distance between \( a_n \) and \( L \) is less than \( \epsilon \). This means the terms \( a_n \) are getting closer and closer to \( L \) as \( n \) increases.

In practice, a sequence that converges does not necessarily mean the individual terms ever reach \( L \), but rather that they get arbitrarily close. The concept of a limit is a foundation of calculus, providing a way to talk about the behavior of sequences and functions as they tend towards infinity or any particular value.

When proving that a subsequence \( \langle a_{f(n)} \rangle \) shares the same limit as its parent sequence \( \langle a_n \rangle \), we apply the definition of a convergent sequence:

• Firstly, recognize that \( \langle a_n \rangle \) converges to a limit \( L \). This indicates that for every \( \epsilon > 0 \), there exists a point \( N \) such that for all \( n \geq N \), \( |a_n - L| < \epsilon \).
• Since the subsequence \( \langle a_{f(n)} \rangle \) is part of \( \langle a_n \rangle \), for each \( f(n) \) fulfilling \( n \geq N \), it ensures \( f(n) \geq N \) holds as well.
• This means \( |a_{f(n)} - L| < \epsilon \) for all elements in the subsequence where \( f(n) \geq N \).

By applying these steps, the subsequence \( \langle a_{f(n)} \rangle \) is shown to converge to the same limit \( L \) as \( \langle a_n \rangle \), thereby completing the proof and demonstrating that the convergence is preserved in the subsequence.