If $x_k \to L$ then $\sigma_n \to L$: convergence implies $(C,1)$-summability to the same value

Given: A sequence $(x_k)$ of reals converging to $L := \lim_k x_k$, and its Cesaro means $\sigma_n = (n+1)^{-1}\sum_{k=0}^{n} x_k$.

The Cesaro means and $(C,1)$-summability (The Cesaro means $\sigma_n = (x_0 + \dots + x_n)/(n+1)$ and $(C,1)$-summability); $(n+1)\cdot 1_{\mathbb{R}} > 0$ for every $n \in \mathbb{N}$ (Canonical naturals are positive and strictly increasing).

Finite sums (Finite sums and finite products, by recursion) and their laws: additivity, scaling with $\sum_{k<d}\lambda = d\lambda$, splitting $\sum_{k<n} = \sum_{k<m} + \sum_{k=m}^{n-1}$ for $m \le n$, and monotonicity of $\sum$ in its terms (Laws of finite sums and finite products).

Triangle inequality for finite sums: $\big|\sum_{k<n} a_k\big| \le \sum_{k<n}|a_k|$ (Triangle inequality for finite sums).

Convergence: for every rational $\varepsilon > 0$ there is $K$ with $|x_k - L| < \varepsilon$ for all $k \ge K$, and equivalently for every real $\varepsilon > 0$, since below every positive real lies a positive rational (Limits and Cauchy sequences of reals, Sequences of reals: bounded, eventually, frequently, tails, subsequences, The rationals embed densely in the reals).

Reciprocal Archimedean property: for every real $\varepsilon > 0$ there is a natural $m \ge 1$ with $1/m < \varepsilon$ (For every $\varepsilon > 0$ in a complete ordered field there is a natural $n \ge 1$ with $1/n < \varepsilon$).

Order arithmetic: $a > 0$ gives $a^{-1} > 0$ and $0 < a < b$ gives $0 < b^{-1} < a^{-1}$ (Inverses of positives are positive, and reciprocation reverses order); for $c > 0$, $a \le b$ gives $ac \le bc$ (Sign rules for products and monotonicity of multiplication); adding a constant preserves the order and inequalities add (Order is preserved by adding a constant and by adding inequalities); the order is total and transitive (Complete ordered field (least-upper-bound property), Ordered field); $|u| \ge 0$ (Basic properties of the absolute value); and $d \cdot 1_{\mathbb{R}} \le e \cdot 1_{\mathbb{R}}$ whenever $d \le e$ in $\mathbb{N}$ (Canonical naturals are positive and strictly increasing). In each clause above, Sign rules for products and monotonicity of multiplication and Order is preserved by adding a constant and by adding inequalities state the STRICT forms and only those; the nonstrict forms used below are those together with the equality cases, which trichotomy settles, the order being total (Ordered field).

The order on $\mathbb{N}$ is total, so two indices have a larger one ($\le$ is a linear order on $\mathbb{N}$).