Constant Sequences and Index-Shifted Sequences of Real Numbers

Let $\mathbb{R}$ be the real numbers, with the order of their ordered field structure and the absolute value $|\cdot|$, let $\mathbb{N}$ be the set of natural numbers, and let convergence of a sequence of real numbers be as in Limit of a Sequence of Real Numbers.

Let $(a_n)_{n\in\mathbb{N}}$ be a sequence of real numbers and let $c,L\in\mathbb{R}$. Then the following hold.

1. (Constant sequences) (i) The sequence $(c)_{n\in\mathbb{N}}$ all of whose terms equal $c$ converges to $c$.

2. (Index shift) (ii) If $(a_n)$ converges to $L$ and $(b_n)_{n\in\mathbb{N}}$ is given by $b_n=a_{n+1}$, then $(b_n)$ converges to $L$.