Sequences and Their Limits

Analysis is the study of limits, and the simplest is the limit of a sequence of real numbers. Continuity, the derivative, the integral, and the interchange of limit operations all reduce to the single $ϵ$-$M$ definition below.

Sequences

The subscript notation resembles set notation, but a sequence and a set are different objects. A sequence is ordered and may repeat values; the set of values it takes, its range, forgets both. For example $\{(-1{)}^n{\}}_{n=1}^{\infty }$ is the sequence $-1,1,-1,1,\dots$, an infinite list, whereas its range is the two-element set $\{-1,1\}$.

A sequence is usually specified by a formula for its $n$th term. Three that recur throughout the subject:

• $\{1/n{\}}_{n=1}^{\infty }=1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots$, decreasing toward $0$.
• $\{n{\}}_{n=1}^{\infty }=1,2,3,\dots$, growing without bound.
• The constant sequence $\{c{\}}_{n=1}^{\infty }=c,c,c,\dots$ for a fixed $c\in \mathbb{R}$.

The sequence $\{1/n\}$ is bounded ($B=1$ works), and $\{n\}$ is not. This lesson uses the absolute value and its triangle inequality from the foundations constantly; the estimate $|a+b|\le |a|+|b|$ appears in nearly every proof below.

Convergence

The idea is that a sequence converges to $x$ if its terms are eventually as close to $x$ as we like. As close as we like is made precise by a tolerance $ϵ>0$; eventually is made precise by an index $M$ past which the tolerance holds.

The order of the quantifiers is the whole content of the definition. The tolerance $ϵ$ is given first, and only then do we produce $M$, so $M$ is allowed to depend on $ϵ$ — a smaller tolerance generally forces a larger $M$. The condition $|x_n-x|<ϵ$ says $x_n$ lies in the open interval $(x-ϵ,x+ϵ)$. So convergence means: every such interval around $x$, no matter how narrow, contains all but finitely many terms of the sequence.

The last is the standard way to prove divergence: assume a limit, choose one specific $ϵ$ small enough to separate two persistent clusters of terms, and derive a contradiction from the triangle inequality.

The limit is unique

We wrote a limit above because uniqueness must be proved. Once it is, we may write ${\lim }_{n\to \infty }{x}_n=x$ and speak of the limit.

The proof is the prototype of a technique used everywhere in analysis: to show a fixed nonnegative quantity is zero, bound it above by an arbitrary $ϵ>0$.

The maneuver of splitting $y-x$ as $(x_n-x)-(x_n-y)$ (inserting and subtracting a term the sequence gets close to) is called adding and subtracting inside an absolute value, and it appears in nearly every proof in this module.

Convergent sequences are bounded

The converse fails: $\{(-1{)}^n\}$ is bounded but divergent. Boundedness is necessary for convergence, not sufficient; the missing hypothesis is supplied by monotonicity or by the extraction of a convergent subsequence.

Convergence depends only on the tail

Changing, deleting, or inserting finitely many terms cannot affect whether a sequence converges or what it converges to.

The limit does not depend on how the sequence begins. This licenses arguments that establish a property for large $n$ — for instance, a sequence that is monotone only from some index onward converges as soon as the tail is monotone and bounded, because the tail governs the limit.

Divergence to infinity

Some divergent sequences have terms that increase past every bound. It is convenient to record this behavior with its own notation, even though such sequences do not converge to a real number.

The structure mirrors ordinary convergence: an arbitrary threshold $K$ is given first, then an index $M$ past which every term clears it. With this convention,

A sequence diverging to $\pm \infty$ is unbounded, but the converse fails: $x_n=0$ for odd $n$ and $x_n=n$ for even $n$ is unbounded yet does not diverge to $\infty$, because it does not eventually stay above every threshold. The finer bookkeeping for such sequences uses the limit superior and limit inferior.

Summary

Convergence is an $ϵ$-then-$M$ statement; a limit, once shown unique, is written ${\lim }_{n\to \infty }{x}_n$; convergence forces boundedness but not conversely; and only the tail matters.