Convergent Sequences of Real Numbers

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 sequence in $\mathbb{R}$ and $L\in\mathbb{R}$.

¶ $(a_{n})$ converges to $L$, written $a_{n}\to L$, if for every real $\varepsilon>0$ there is $N\in\mathbb{N}$ with $|a_{n}-L|<\varepsilon$ for every $n\in\mathbb{N}$ with $n\ge N$. It is convergent if it converges to some real number, and a null sequence if it converges to $0$.