Final Answer:
If a sequence aₙ is convergent with limit L, any subsequence of aₙ is also convergent with the same limit L.
Step-by-step explanation:
To prove this statement, let's consider a sequence aₙ that converges to a limit L. Mathematically, this is expressed as limₙ→∞ aₙ = L. Now, let's take any subsequence aₙₖ of aₙ, where nₖ is a strictly increasing sequence of indices. We want to show that aₙₖ also converges to L.
By the definition of a convergent sequence, for any given ε > 0, there exists a positive integer N such that for all n ≥ N, |aₙ - L| < ε. Now, consider the subsequence aₙₖ. Since nₖ is strictly increasing, we have nₖ ≥ k. Therefore, for all k ≥ N, we have |aₙₖ - L| < ε, satisfying the definition of convergence. This establishes that any subsequence aₙₖ of aₙ converges to the same limit L.
This property is fundamental in real analysis and provides a useful tool for analyzing the behavior of subsequences in relation to the convergence of the original sequence. It underscores the connection between the convergence of a sequence and the convergence of its subsequences, emphasizing the coherence in the behavior of the elements in the sequence.