Proof that a series of nonnegative terms converges iff its partial sums are bounded

Builds on Convergent series Partial sum Monotone convergence theorem for sequences Convergent sequences are bounded Monotone sequence Sum of a series

Proof

Let $s_N = \sum_{n=0}^{N} a_n$ be the partial sums. Since $s_{N+1} - s_N = a_{N+1} \ge 0$ , the sequence $(s_N)$ is increasing, as a monotone sequence.
Assume that $(s_N)$ is bounded above. By the monotone convergence theorem, $(s_N)$ converges, and its limit is $\sup_N s_N$ .
By the definition of a convergent series, $\sum a_n$ converges. Its sum is the limit of $(s_N)$ , which is $\sup_N s_N$ by step 2.
Conversely, assume that $\sum a_n$ converges. Then $(s_N)$ is a convergent sequence.
Convergent sequences are bounded, so $(s_N)$ is bounded, and in particular bounded above.
Steps 2 to 5 prove the equivalence. When it holds, step 3 gives $\sum_{n=0}^{\infty} a_n = \sup_N s_N$ .

Idea

Nonnegative terms make the partial sums increase. An increasing sequence has only one way to fail to converge: it grows without bound. So boundedness is exactly what is needed.

Reading path

To understand this page from the ground up, read these 54 pages in order. The foundations come first. The path takes the proofs Proof of the approximation property of the supremum, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof that convergent sequences are bounded, Proof of the monotone convergence theorem for sequences.

Construction of the reals from Cauchy sequences concept
Element definition
Integers object
Mathematical induction concept
Ordered field definition
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Set definition
Ring of integers statement
Implication definition
Predicate definition
The reals form a field statement
Real sequence definition
Supremum definition
Ordered pair definition
Universal quantifier definition
The rationals form a field statement
Bounded sequence definition
Limit of a sequence definition
Monotone sequence definition
Series definition
Completeness axiom statement
Cartesian product definition
Subset definition
The rationals form an ordered field statement
Convergent sequence definition
Partial sum definition
Binary relation definition
Order on the reals definition
Convergent series definition
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Sum of a series definition
Preorder definition
Partial order definition
Total order definition
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
The reals form an ordered field statement
Absolute value definition
Proof of the approximation property of the supremum proof
Proof of the triangle inequality for real numbers proof
Proof of the epsilon-N definition of the limit of a sequence proof
Approximation property of the supremum statement
Triangle inequality statement
Epsilon-N definition of the limit of a sequence statement
Proof that convergent sequences are bounded proof
Proof of the monotone convergence theorem for sequences proof
Convergent sequences are bounded statement
Monotone convergence theorem for sequences statement
Proof that a series of nonnegative terms converges iff its partial sums are bounded proof

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Tag	`XV374C0RVT`
URI	`m2://XV374C0RVT`
Kind	proof
Writing	W0 stub
MSC	40A05
Proves	A series of nonnegative terms converges iff its partial sums are bounded
Uses	Convergent series, Partial sum, Monotone convergence theorem for sequences, Convergent sequences are bounded, Monotone sequence, Sum of a series
Changed	4 November 2026

Proof that a series of nonnegative terms converges iff its partial sums are bounded

Proof

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