35 Convergence of sequences
Lemma 35.1 (Toeplitz Lemma) Let \(\{a_n\}_{n \ge 0}\) be a sequence of non-negative numbers and \(b_n = \sum_{k=0}^n\) is such that \(b_n > 0\) for all \(n \ge 0\) and \(\lim_{n \to ∞} b_n = ∞\). Let \(\{x_n\}_{n \ge 0}\) be a sequence of numbers that converge to \(x\). Then \[ \lim_{n \to ∞} \frac{1}{b_n} \sum_{k=0}^{n-1} a_k x_k = x. \]
If we take \(a_n = 1\), Lemma 35.1 reduces to Ceasaro’s mean.
Lemma 35.2 (Kronecker Lemma) Let \(\{b_n\}_{n \ge 0}\) be a seuquence of positive numbers such that \(\lim_{n \to ∞} b_n = \infty\). Let \(\{x_n\}_{n \ge 0}\) be a sequence of numbers such that \(\sum_{n=0}^{∞} x_n < \infty\). Then, \[ \lim_{n \to ∞} \frac 1{b_n} \sum_{j=1}^n b_j x_j = 0. \]