Proof of the comparison test

Builds on A series of nonnegative terms converges iff its partial sums are bounded Partial sum

Proof

Let $A_N = \sum_{n=0}^{N} a_n$ and $B_N = \sum_{n=0}^{N} b_n$ be the partial sums. Since $a_n \le b_n$ for every $n$ , we have $A_N \le B_N$ for every $N$ .
Assume that $\sum b_n$ converges, with sum $B$ . Since $b_n \ge 0$ , the criterion for nonnegative series gives $B = \sup_N B_N$ . So $B_N \le B$ for all $N$ .
By steps 1 and 2, $A_N \le B$ for all $N$ . So the partial sums of $\sum a_n$ are bounded above.
Since $a_n \ge 0$ , the same criterion shows that $\sum a_n$ converges, with sum $\sup_N A_N$ .
By step 3, $B$ is an upper bound of the $A_N$ . So $\sum a_n = \sup_N A_N \le B = \sum b_n$ .
Now assume that $\sum a_n$ diverges. If $\sum b_n$ converged, then $\sum a_n$ would converge by steps 2 to 4. So $\sum b_n$ diverges.

Idea

The partial sums of the smaller series stay below those of the larger one. For nonnegative terms, convergence means bounded partial sums, so a bound for the larger series is also a bound for the smaller one.

Reading path

To understand this page from the ground up, read these 56 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, Proof that a series of nonnegative terms converges iff its partial sums are bounded.

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
A series of nonnegative terms converges iff its partial sums are bounded statement
Proof of the comparison test proof

Where this leads

671 pages build on this one.

Tag	`JVHCSNBVGQ`
URI	`m2://JVHCSNBVGQ`
Kind	proof
Writing	W0 stub
MSC	40A05
Proves	Comparison test
Uses	A series of nonnegative terms converges iff its partial sums are bounded, Partial sum
Changed	4 November 2026

Proof of the comparison test

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