= Limit of a sequence
{title2=$\lim_{n\to\infty}x_n$}
= Limits of sequences
{synonym}
A <sequence> $(x_n)_{n\geq1}$ of <real numbers> has <limit of a sequence> $L\in\mathbb R$, written $x_n\to L$, precisely when
$$
\forall\varepsilon>0\;\exists N\in\mathbb N_{>0}\;\forall n\geq N:\quad |x_n-L|<\varepsilon.
$$
The <absolute value> measures the error. The tolerance $\varepsilon$ is chosen first; $N$ may depend on it, but must control every later term. Infinitely many accurate terms alone do not suffice. For example, $x_n=(-1)^n$ repeatedly equals one but also repeatedly differs from one by two. The definition and uniqueness of the <limit of a sequence> are discussed in https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/mit18_100af20_lec62.pdf[MIT 18.100A, Lecture 6, definitions 13 and theorem 15].
For a worked error estimate, let $x_n=3+2/n$. Then $|x_n-3|=2/n$, so any positive integer $N>2/\varepsilon$ works. With $\varepsilon=0.01$, choose $N=201$; every later error is at most $2/201<0.01$. The strict inequality matters: $N=200$ would give equality at its first term. A <limit of a sequence> turns the informal instruction “take enough terms” into an explicit accuracy guarantee.
The <limit of a sequence> is unique: two distinct proposed answers cannot both be within arbitrarily small error of every late term, by the <triangle inequality>. Changing finitely many initial terms does not change it. For an <extended real number>, $x_n\to+\infty$ means $\forall M\in\mathbb R\;\exists N\;\forall n\geq N:\ x_n>M$; convergence to $-\infty$ similarly requires $x_n<M$. These describe escape rather than a finite answer. The <extended-real Fekete lemma> is an example where the <limit of a sequence> need not be finite.
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