Part 2/5:
We start by recalling the definition of a limit for sequences. A sequence (a_n) has a limit (L) if, for every (\epsilon > 0), there exists an integer (N) such that for all integers (n > N), the absolute difference (|a_n - L| < \epsilon). This definition emphasizes that as (n) increases, the terms of the sequence get arbitrarily close to the limit (L).
Establishing the Limit Relationship
By replacing (n) with (n + 1) in the definition, we assert that if (n > N), then (|a_{n+1} - L| < \epsilon). This rephrasing indicates that the limit of (a_{n+1}) also converges to (L) as (n) approaches infinity.
Thus, we reach the conclusion:
[
\lim_{n \to \infty} a_{n+1} = L
]