Math 2.0

Sequences, series, summability Proposition

Limit of a product of sequences

Builds on Limit of a sequenceConvergent sequences are bounded

Statement

Let (an)(a_n) and (bn)(b_n) be real sequences, and let aa and bb be real numbers. If an→aa_n \to a and bn→bb_n \to b as n→∞n \to \infty, then anbn→aba_n b_n \to a b as n→∞n \to \infty.

Remarks

The proof uses the identity anbn−ab=an(bn−b)+b(an−a)a_n b_n - a b = a_n (b_n - b) + b (a_n - a). The sequence (an)(a_n) is bounded, because convergent sequences are bounded. With a constant sequence an=ca_n = c, it gives c bn→c bc\, b_n \to c\, b. By induction, it gives ank→aka_n^k \to a^k for each natural number kk.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionEpsilon-N definition of the limit of a sequence, statement, no evidence levelEpsilon-N definition …statementConvergent sequences are bounded, statement, no evidence levelConvergent sequences …statementLimit of a product of sequences, statement, no evidence levelLimit of a product of…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Sequence definition
  2. Real sequence definition
  3. Bounded sequence definition
  4. Limit of a sequence definition
  5. Convergent sequence definition
  6. Epsilon-N definition of the limit of a sequence statement
  7. Convergent sequences are bounded statement
  8. Limit of a product of sequences statement

Where this leads

17 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Limit_of_a_sequence, Properties
About this page
Tag0734RSVF2V
URIm2://0734RSVF2V
Kindstatement, proposition
WritingW0 stub
MSC40A05
StatusThis copy of the site has no build data. Run m2 status 0734RSVF2V.
UsesLimit of a sequence, Convergent sequences are bounded

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