Limit of a Sequence¶

Definition¶

Let $L\in\mathbb{R}\text{.}$ A sequence $(a_n)_{n=1}^\infty$ converges to $L$ if for every $\varepsilon >0\text{,}$ there exists an $N \geq 1$ such that for every $n\geq N\text{,}$ we have $|a_n-L|<\varepsilon\text{.}$ In this case, the sequence $(a_n)_{n=1}^\infty$ is called convergent, and $L$ is called its limit. We write

If no such $L\in\mathbb{R}$ exists, the sequence $(a_n)_{n=1}^\infty$ is called divergent.

Interpretation¶

The limit of a sequence describes the value approached by its terms as the index increases. The definition formalizes "approached" as eventual closeness: however small a tolerance $\varepsilon>0$ is chosen, from some index $N$ on every term lies within $\varepsilon$ of $L$.

$a_n=1/n\to0$$0$: for $\varepsilon=0.1$$0.1$, every term with $n\ge N=11$ (red) lies within $\varepsilon$ of the limit $0$, since $1/n<0.1$ exactly when $n>10$$10$. A smaller $\varepsilon$ needs a larger $N$: for $\varepsilon=0.01$$0.01$, $N=101$ works. The definition asks only that some $N$ exists for each $\varepsilon$, not the smallest one.

The order of the quantifiers carries the meaning: first $\varepsilon$ is given, then $N$ may be chosen depending on $\varepsilon$, and the condition must hold for all $n\ge N$$N$. This last part rules out sequences that only come close now and then. The terms of $(-1)^n$ equal $1$ infinitely often but also return to $-1$, so for $\varepsilon=1$ no $N$ keeps all later terms within $\varepsilon$ of any candidate $L$. Only the tail matters: changing finitely many terms does not change whether the limit exists or what it is.

Two slips are common. "The terms get closer and closer to $L$" is not the definition: the terms of $1/n$ also get closer to $-1$ at every step, but $-1$ is not the limit. And the terms need not approach from one side: $(-1)^n/n$ converges to $0$ while jumping from one side of $0$ to the other. The same $\varepsilon$-idea returns later for limits of functions, as a limit at a point.

Key Results¶

• Limits of sequences are unique when they exist.
• Every convergent sequence is bounded.

Examples¶

• $\displaystyle \lim_{n\to\infty}\frac{1}{n}=0\text{.}$
• The sequence $(-1)^n$ does not converge.

Prerequisites¶

• Sequence

Related Concepts¶

• Cauchy Sequence
• Bounded Sequence
• Limit at Infinity
• Monotone Sequence
• Subsequence
• Real Number

Sources¶

• Math I: Calculus · Lecture 3

Aliases: sequence limit Keywords: convergent sequence, convergence of a sequence Symbols: element of, epsilon, infinity, real numbers

Last updated: 27 October 2026 11:52