I am always confused with these type of terms. Consider the following definition of convergence of sequences and the example of how sometimes is used. This proof appears in Stromberg.
Definition: A sequence $(x_n)_{n=1}^\infty \subset\mathbb{C}$ is said to converge if there exists $x\in\mathbb{C}$ such that to each $\epsilon >0$ corresponds $n_\epsilon \in \mathbb{N}$ for which $$|x-x_n|<\epsilon$$ for all $n\geq n_\epsilon$.
Theorem: Let $(\alpha_n)_{n=0}^\infty$ and $(\beta_n)_{n=0}^\infty$ be sequences in $\mathbb{C}$ and let $\alpha, \beta \in\mathbb{C}$. If $\alpha_n \rightarrow \alpha$, then $$\frac{1}{n+1}\sum_{k=0}^n \alpha_k \rightarrow \alpha.$$
Let $\epsilon >0$ be given. Choose $n_0\in \mathbb{N}$ such that $|\alpha - \alpha_k|<\epsilon/2$ for all $k>n_0$. Next choose $n_1>n_0$ such that $\frac{1}{n+1}\sum_{k=0}^{n_0}<\epsilon/2$ for all $n>n_1$. Then, $n> n_1$ implies
\begin{equation} \begin{split} \left|\alpha - \frac{1}{n+1}\sum_{k=0}^n \alpha_k \right| &= \left|\frac{1}{n+1}\sum_{k=0}^n (\alpha - \alpha_k) \right|\\ &\leq \frac{1}{n+1}\sum_{k=1}^{n_0}|\alpha - \alpha_k| + \frac{1}{n+1}\sum_{k=n_0 +1}^n |\alpha - \alpha_k|\\ &<\frac{\epsilon}{2} + \frac{n-n_0}{n+1}\frac{\epsilon}{2}<\epsilon, \end{split} \end{equation}
and so it is proven the statement.
Why the author from the outset chooses $k>n_0$ and does not stick to $k\geq n_0$? Is it because of how he separates the sums? Also, is it really necessary to request $n_1>n_0$? Is it not more simply to request $ n\geq \max\lbrace n_0,n_1\rbrace$?