Final Answer:
If a sequence (aₙ) is convergent with limit (L), then any subsequence of (aₙ) will also converge to the same limit (L).
Step-by-step explanation:
Convergence of a sequence implies that as (n) approaches infinity, the terms of the sequence get arbitrarily close to a specific value, which is the limit (L). When considering a subsequence, it consists of terms selected from the original sequence in the order of natural numbers. Since the original sequence converges to (L), any subset of terms, arranged in a subsequence, will still approach the same limit (L).
Mathematically, let (aₙ) be the given convergent sequence with limit (L), and let (bₖ) be a subsequence of (aₙ) indexed by natural numbers (k). The convergence of (aₙ) to (L) implies that for any positive real number ε, there exists a positive integer (N) such that for all (n > N), (|aₙ - L| < ε). Since the terms of the subsequence (bₖ) are selected from the terms of (aₙ), this same property holds for the subsequence, and for all (k > N), (|bₖ - L| < ε).
Therefore, the convergence behavior is inherited by any subsequence, and they converge to the same limit as the original sequence. This is a fundamental property of convergent sequences and subsequences, providing insight into the consistent behavior of terms as the index increases indefinitely.