 Limit of a sequence

A sequence $(x_n{)}_{n\ge 1}$ of real numbers has limit of a sequence $L\in \mathbb{R}$, written $x_n\to L$, precisely whenThe absolute value measures the error. The tolerance $\epsilon$ 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 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 $N>2/\epsilon$ works. With $\epsilon =0.01$, 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 $\forall M\in \mathbb{R}\exists N\forall n\ge N:x_n>M$; 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.

The limit of a sequence refers to the value that the terms of the sequence approach as the index (usually denoted as \( n \)) goes to infinity.