In the world of sequences and series, one of the places of interest is the bounded sequence. Not all sequences are bonded. In this article, you will learn which sequences are bonded and how they are bonded.
| Term | Meaning |
|---|---|
| Monotonic increasing | a_(n+1) ≥ a_n for all n |
| Monotonic decreasing | a_(n+1) ≤ a_n for all n |
| Bounded above | ∃ K such that a_n ≤ K for all n |
| Bounded below | ∃ k such that a_n ≥ k for all n |
| Bounded | Both bounded above and bounded below |
| Supremum (sup) | The smallest upper bound |
| Infimum (inf) | The greatest lower bound |
| Monotone Convergence Theorem | Monotonic + bounded ⟹ convergent |
Monotonic vs Non-Monotonic Sequences
Before defining bounded sequences, we need to understand the distinction between monotonic and non-monotonic sequences, since this affects how we analyse their bounds.
Monotonic sequences
A sequence is monotonically increasing if each term is greater than or equal to the one before it, and monotonically decreasing if each term is less than or equal to the one before it. Consider the sequence:
The terms are always increasing, and they appear to approach 1 without ever reaching it. This is a monotonically increasing sequence — and crucially, it is bounded.
Non-monotonic sequences
Consider the alternating sequence:
The sign alternates, so this sequence is not monotonic. However — and this is a common misconception — a non-monotonic sequence can still be bounded. Here, every term lies between 
and 
, so the sequence is perfectly well bounded.
"Non-monotonic" does not mean "unbounded." A sequence can oscillate and still have a clear upper and lower bound. These are independent properties.
What is a Bounded Sequence?
Intuitively, a sequence is bounded if all its terms are trapped between two finite values — no matter how far along the sequence you go, no term escapes beyond a ceiling or a floor.
Bounded Sequence Definition:
A sequence 
is bounded if there exist real numbers 
and 
such that:
The value is a lower bound and is an upper bound. Note that the bounds themselves don't need to be the tightest possible values — any and that work will do. However, mathematicians are often interested in the tightest bounds, which leads us to the concepts of infimum and supremum.
Bounded Above, Below, and the Infimum/Supremum
Definition — Bounded Below A sequence is bounded below if there exists a real number k such that a_n ≥ k for all n. The value k is called a lower bound. The greatest lower bound — the largest value of k that still satisfies this inequality — is called the infimum, written 
.
A sequence is bounded above if there exists a real number K such that a_n ≤ K for all n. The value K is called an upper bound.
The smallest upper bound — the smallest value of K that still satisfies this inequality — is called the supremum, written 
.
💡 Exam Tip: The infimum and supremum may or may not actually be terms of the sequence. For example, in the sequence:
The supremum is 1 — but no term ever equals 1. When the supremum is achieved, it is also the maximum. When it is not achieved, it is the least upper bound that the sequence approaches but never reaches.
Connection to Convergence
Boundedness and convergence are closely linked. Here is one of the most important theorems you need to know
Every sequence that is both monotonic and bounded is convergent.
A monotonically increasing sequence that is bounded above converges to its supremum.
A monotonically decreasing sequence that is bounded below converges to its infimum.
Applying this to our examples: the sequence 
is monotonically increasing and bounded above by 1. The Monotone Convergence Theorem guarantees it converges — and indeed 
.
⚠️Important Caveat: The converse is not always true: a convergent sequence is always bounded, but a bounded sequence is not necessarily convergent. For example, (-1)^n is bounded but does not converge.
The Monotone Convergence Theorem is frequently used in proofs. If you can show a sequence is monotone and bounded, you immediately know it converges — even if you cannot find the limit explicitly.
Worked Examples
Determine whether this sequence is bounded:
Step 1: Write out the first few terms to understand the behaviour.
The sequence is decreasing and positive.
n: 
1: 3
2: 3/2
3: 1
4: 3/4
10: 3/10
→ ∞: → 0
Step 2: Find the supremum (upper bound). Since the sequence is decreasing, its largest term is the first: a₁ = 3. Therefore, 3 is the supremum. Every term satisfies 
.
Step 3: Find the infimum using the limit as n → ∞:
lim (n→∞) 3/n = 0
The sequence approaches 0 but never reaches it. Therefore, 0 is the infimum. Every term satisfies 
.
∴ The sequence is bounded: 0 < a_n ≤ 3 for all n ∈ ℕ. Infimum = 0, Supremum = 3.
Determine whether the sequence is bounded.
Step 1: Write out the first few terms.
n:
1: 1/2
2: 2/3
3: 3/4
4: 4/5
10: 10/11
→ ∞: → 1
The sequence is increasing and always less than 1.
Step 2: Find the infimum. Since the sequence is increasing, its smallest term is the first: b₁ = 1/2. Therefore, 1/2 is the infimum. Every term satisfies 
.
Step 3: Find the supremum using the limit as n → ∞:
lim (n→∞) n/(n+1) = lim (n→∞) 1 / (1 + 1/n) = 1
The sequence approaches 1 but never reaches it. Therefore, 1 is the supremum. Since terms get arbitrarily close to 1, no smaller value can serve as the upper bound.
∴ The sequence is bounded: 1/2 ≤ b_n < 1 for all n ∈ ℕ. Infimum = 1/2, Supremum = 1.
Determine whether the sequence 
is bounded.
Step 1: Write out the first few terms.
n:
1: -1
2: 1/2
3: -1/3
4: 1/4
5= -1/5
The sequence alternates in sign, and its magnitude decreases towards 0.
Step 2: Establish bounds using the absolute value. For all n, we have |c_n| = 1/n ≤ 1, so -1 ≤ c_n ≤ 1/2. The lowest term is c₁ = -1 (infimum, achieved) and the highest is c₂ = 1/2 (supremum, achieved).
∴ The sequence is bounded despite being non-monotonic. Infimum = -1, Supremum = 1/2.
Summarise with AI:
Did you like this article? Rate it!
Loading...
