Sequence Convergence and Divergence
Summarize
Convergent sequence
Sequences are ordered collections of elements, each assigned to a specific position indexed by a natural number. For every sequence $(a_n)_{n \in \mathbb{N}}$, there is an associated behavior of its terms $a_n$ that describes how they evolve as the index $n$ increases.
Analyzing this behavior helps determine whether the sequence converges to a finite limit, diverges to infinity, or exhibits an oscillating pattern.
Definition 1. A sequence $(a_n)_{n \in \mathbb{N}}$ is said to be convergent to the limit $\ell \in \mathbb{R}$ if for every $\varepsilon > 0$, there exists $n_0 \in \mathbb{N}$ such that:
$$ |a_n - \ell| < \varepsilon \quad \text{for all } n \geq n_0. $$
In this case, we write:
$$ \lim_{n \to +\infty} a_n = \ell \quad \text{or} \quad a_n \to \ell \quad \text{as } n \to +\infty. $$
In other words, this means that the terms of the sequence get increasingly close to the number $\ell$ as $n$ grows larger. No matter how tight a margin $\varepsilon$, from a certain index onward all terms will stay within that distance from $\ell$. For example, consider the following sequence: $$ a_n = \left( \frac{1}{n} \right)_{n \geq 1} = \left(1, \frac{1}{2}, \frac{1}{3}, \ldots \right) $$
As $n$ increases, the terms become smaller and smaller, approaching zero.
This is a classic example of a sequence that converges to 0. A sequence is said to be infinitesimal when its terms get arbitrarily close to zero as the index grows and:
$$ \lim_{n \to +\infty} a_n = 0. $$
The limit of a sequence $(a_n)_{n \in \mathbb{N}}$, if it exists, is unique. Suppose, by contradiction, that the sequence converges both to $\ell_1$ and to $\ell_2$ with $\ell_1 \neq \ell_2$.
Choosing $\varepsilon = |\ell_1 - \ell_2|/2 > 0$, the convergence to $\ell_1$ gives an index $n_1$ such that $|a_n - \ell_1| < \varepsilon$ for $n \geq n_1$, and the convergence to $\ell_2$ gives an index $n_2$ with $|a_n - \ell_2| < \varepsilon$ for $n \geq n_2$.
For $n \geq \max\{n_1, n_2\}$ the triangle inequality yields a contradiction:
$$|\ell_1 - \ell_2| \leq |\ell_1 - a_n| + |a_n - \ell_2| < 2\varepsilon = |\ell_1 - \ell_2|$$
So, he two candidate limits must therefore coincide.
Example
Let's consider the sequence defined by:
$$ a_n = \frac{n}{n + 2} $$
We aim to demonstrate that this sequence converges to 1 as $n \to +\infty$, using the formal definition of convergence.
To prove this, we must show that for every $\varepsilon > 0$, there exists a natural number $n_0$ such that for all $n \geq n_0$:
$$ \left| \frac{n}{n + 2} - 1 \right| < \varepsilon $$
Let's simplify the absolute value expression:
$$ \left| \frac{n}{n + 2} - 1 \right| = \left| \frac{-2}{n + 2} \right| = \frac{2}{n + 2}. $$
Now, we want:
$$ \frac{2}{n + 2} < \varepsilon $$
Solving the inequality:
$$ n + 2 > \frac{2}{\varepsilon} \quad \Rightarrow \quad n > \frac{2}{\varepsilon} - 2 $$
So we can define:
$$ n_0 = \left\lceil \frac{2}{\varepsilon} - 2 \right\rceil $$
From this point onward, every term of the sequence stays within a distance $\varepsilon$ of the limit $1.$ Hence, by definition:
$$ \lim_{n \to +\infty} \frac{n}{n + 2} = 1. $$
Divergent sequence
A sequence $(a_n)_{n \in \mathbb{N}}$ is said to be divergent if it does not converge to a finite limit. This can happen in the following ways.
A sequence diverges to $+\infty$ if, for every $M > 0$, there exists an index $n_0 \in \mathbb{N}$ such that
$$
a_n > M \quad \forall \ n \geq n_0
$$
In this case, we write:
$$
\lim_{n \to +\infty} a_n = +\infty \quad \text{or} \quad a_n \to +\infty \text{ as } n \to +\infty
$$
A sequence diverges to $-\infty$ if, for every $M < 0$, there exists an index $n_0 \in \mathbb{N}$ such that
$$
a_n < M \quad \forall \ n \geq n_0
$$
In this case, we write:
$$
\lim_{n \to +\infty} a_n = -\infty \quad \text{or} \quad a_n \to -\infty \text{ as } n \to +\infty
$$
Bounded sequence
A bounded sequence is a sequence of numbers whose terms always stay within a fixed, finite interval, no matter how large the index becomes. In formal terms, let $\{a_n\}$ be a sequence. We say that the sequence is bounded if there exists a constant $M > 0$ such that:
$$ |a_n| \leq M \quad \forall n \in \mathbb{N} $$
We say that a sequence $\{a_n\}$ is bounded above if there exists a constant $M \in \mathbb{R}$ such that:
$$ a_n \leq M \quad \forall n \in \mathbb{N} $$
We say that the sequence is bounded below if there exists a constant $M \in \mathbb{R}$ such that:
$$ a_n \geq M \quad \forall n \in \mathbb{N} $$
Oscillating sequence
Oscillating sequences are a special type of bounded sequence. Let us consider the sequence:
$$ (a_n)_{n \in \mathbb{N}} = ((-1)^n)_{n \in \mathbb{N}} = (+1, -1, +1, -1, +1, -1, \dots) $$
As the index $n$ increases, the terms of the sequence alternate consistently between $+1$ and $-1$.
This type of sequence does not approach any finite value and is called an oscillating sequence. It does not converge to a finite limit, nor does it diverge to $+\infty$ or $-\infty$, and its terms continue to fluctuate between different values. The extremal values around which the terms cluster for arbitrarily large indices are captured precisely by the superior and inferior limits of the sequence.
Geometric sequence
Let us consider an example of a sequence, called a geometric sequence, which can display different behaviors depending on the fixed real number $q$. In general, a numerical sequence is called a geometric progression when the ratio between each term and its previous one is constant. More precisely, a geometric sequence is defined as follows:
$$ a_n := q^n $$
The behavior of the sequence depends entirely on the value of $q$:
- It diverges to $+\infty$ if $q > 1$, since the powers $q^n$ grow without bound.
- It is constant and equal to $1$ if $q = 1$, since $a_n = 1^n = 1$ for every $n$.
- It is infinitesimal if $|q| < 1$, that is, for $-1 < q < 1$ with $q \neq 0$: the terms approach zero, alternating in sign when $q$ is negative.
- It is a bounded oscillating sequence if $q = -1$, since $a_n$ alternates between $+1$ and $-1$ and admits no limit.
- It diverges in an oscillatory manner if $q < -1$: the absolute values grow without bound and the sign alternates, so the sequence is unbounded and admits no limit.
As shown in the graph, when $q = 2$, the values of the geometric sequence $a_n = q^n$ grow exponentially. As $n$ increases, each term doubles the previous one, leading to a rapid escalation in magnitude.
Take a closer look at the difference between an arithmetic progression and a geometric progression to better understand how their structures and growth patterns differ.
Algebra of limits
The limit of a sequence interacts in a predictable way with the elementary operations of arithmetic. Let $(a_n)$ and $(b_n)$ be two convergent sequences, with $a_n \to \ell$ and $b_n \to m$. The following identities hold:
- Sum: $\lim_{n \to \infty} (a_n + b_n) = \ell + m$.
- Difference: $\lim_{n \to \infty} (a_n - b_n) = \ell - m$.
- Product: $\lim_{n \to \infty} (a_n \cdot b_n) = \ell \cdot m$.
- Quotient: $\lim_{n \to \infty} \dfrac{a_n}{b_n} = \dfrac{\ell}{m}$, provided $m \neq 0$ and $b_n \neq 0$ from some index onward.
- Constant multiple: $\lim_{n \to \infty} (c \cdot a_n) = c \cdot \ell$ for every $c \in \mathbb{R}$.
- Absolute value: $\lim_{n \to \infty} |a_n| = |\ell|$.
These rules follow directly from the $\varepsilon$-definition of convergence and the triangle inequality. The identity for the sum, for instance, is obtained by splitting $\varepsilon$ into two equal parts: choose indices $n_1, n_2$ such that $|a_n - \ell| < \varepsilon/2$ for $n \geq n_1$ and $|b_n - m| < \varepsilon/2$ for $n \geq n_2$.
Then for $n \geq \max\{n_1, n_2\}$ the triangle inequality gives:
$$ |(a_n + b_n) - (\ell + m)| \leq |a_n - \ell| + |b_n - m| < \varepsilon $$
The other identities admit analogous proofs.
When at least one of the two limits is infinite, the rules above remain valid in extended form whenever the result is unambiguous: for instance, $\ell + \infty = +\infty$ for every finite $\ell$. The configurations that produce no canonical value, such as $\infty - \infty$ or $0 \cdot \infty$, are known as indeterminate forms and require additional analysis.
The squeeze theorem
When a sequence is trapped between two others sequences that both converge to the same limit, then it also converges to that limit. This is the squeeze theorem.
Theorem 1. Let $(a_n)$, $(b_n)$, $(c_n)$ be three sequences such that $a_n \leq b_n \leq c_n$ for every $n$ sufficiently large. If $\lim_{n \to \infty} a_n = \lim_{n \to \infty} c_n = \ell$, then $\lim_{n \to \infty} b_n = \ell$.
The proof is direct. Fix $\varepsilon > 0$. By convergence of $a_n$ and $c_n$ to $\ell$, there exists an index $n_0$ such that for every $n \geq n_0$ both $\ell - \varepsilon < a_n$ and $c_n < \ell + \varepsilon$ hold. The chain $\ell - \varepsilon < a_n \leq b_n \leq c_n < \ell + \varepsilon$ then yields $|b_n - \ell| < \varepsilon$, which is the definition of convergence.
The squeeze theorem is particularly useful when a sequence is too complicated to be handled directly, but admits clear upper and lower bounds. The sequence $b_n = \sin(n)/n$, for example, is squeezed between $-1/n$ and $1/n$, both of which tend to zero; the theorem gives $\lim b_n = 0$ at once.
Sign permanence
If a sequence converges to a strictly positive limit, then its terms are eventually positive as well. More precisely, if $a_n \to \ell$ with $\ell > 0$, then there exists $n_0 \in \mathbb{N}$ such that $a_n > 0$ for every $n \geq n_0$. The symmetric statement for $\ell < 0$ also holds: the terms are eventually negative.
To see this, choose $\varepsilon = \ell/2 > 0$ in the definition of convergence. There exists $n_0$ such that $|a_n - \ell| < \ell/2$ for every $n \geq n_0$, which rearranges to $\ell/2 < a_n < 3\ell/2$. In particular $a_n > \ell/2 > 0$, as claimed.
The hypothesis that the limit is strictly positive cannot be weakened to $\ell \geq 0$: the sequence $a_n = -1/n$ converges to $0$ but every term is negative. Sign permanence is a strict property of strictly positive (or strictly negative) limits.
Subsequences and the Bolzano-Weierstrass theorem
A subsequence of $(a_n)$ is obtained by selecting an increasing collection of indices $n_1 < n_2 < n_3 < \cdots$ and considering the new sequence $(a_{n_k})_{k \in \mathbb{N}}$. Subsequences inherit the convergence of the parent sequence: if $a_n \to \ell$, then every subsequence also converges to $\ell$. This observation provides a frequently used criterion for non-convergence: if two subsequences of $(a_n)$ converge to different limits, then $(a_n)$ does not converge. The superior and inferior limits give a systematic account of this phenomenon, identifying the largest and smallest values that can be obtained as subsequential limits.
The converse question, whether a sequence that fails to converge nevertheless admits a convergent subsequence, is answered by the Bolzano-Weierstrass theorem.
Theorem 2. Every bounded sequence of real numbers admits a convergent subsequence.
The result is a structural property of the real line and follows from completeness. It plays a central role in the theory of Cauchy sequences, where it is used to upgrade boundedness into convergence, and in the proof of Weierstrass' theorem on the attainment of extrema by continuous functions on a compact interval.