#math/analysis Let $a_{1},a_{2},a_{3},a_{4},\ldots=(a_{n}),n\in \mathbb{N}$ be a sequence of real numbers. We say the sequence $(a_{n})$, `converges to the limit`, $b\in \mathbb{R}$ as $n\to \infty$ provided that for ever $\epsilon>0$, there exists some $N\in \mathbb{N}$ such that for all $n\geq{}N$ we have $ \lvert a_{n}-b \rvert <\epsilon. $ In other words, if we think of $n=1,2,\ldots$ as a sequence of times, then the sequence $(a_{n})$ converges to $b$ provided that *eventually* all its terms nearly equal $b$. Utilizing notation from our preliminaries, this is more formally written as $ \forall \:{}\epsilon>0\:{}\exists N\in \mathbb{N}\text{ such that }n\geq{}N\implies \lvert a_{n}-b \rvert <\epsilon. $ When $b$ exists, it is unique, and we write $ \lim_{ n \to \infty } a_{n}=b\quad\text{ or }\quad a_{n}\to b. $