Final answer:
To prove that the arithmetic mean of a sequence converges to the same limit as the sequence itself, we use the definition of limit convergence and properties of limits. The influence of early terms in the sequence diminishes as n grows, leading to the average being dominated by terms close to the limit, thereby converging to the same value.
Step-by-step explanation:
The question is about showing that if a sequence zn converges to a limit γ as n approaches infinity, then the average of the first n terms of the sequence also converges to γ as n approaches infinity. This can be shown using the definition of convergence and the properties of limits.
Firstly, since zn → γ, for every ε > 0, there exists an N such that |zn - γ| < ε for all n > N. Therefore, for n sufficiently large, each term zi contributing to the average [z1 + z2 + ... + zn]/n will be close to γ.
Next, consider the sequence of averages An = (z1 + z2 + ... + zn)/n. As n increases, the influence of any fixed number of terms in the sequence on the average decreases, tending to zero. Thus, the average becomes more influenced by the terms that are close to γ, and hence An will approach γ as n approaches infinity. By applying the arithmetic of limits, we can formally prove this result, which is consistent with the statement we want to prove.