The Limit of a Convergent Sequence

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let $(a_{n})$ be a convergent sequence in $\mathbb{R}$.

¶ The limit of $(a_{n})$, written $\lim_{n\to\infty}a_{n}$, is the real number to which $(a_{n})$ converges; it is unique by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §unique.