◀ ▲ ▶Branches / Analysis / Proposition: Product of a Real Number and a Convergent Real Sequence

applicability: $\mathbb {N, Z, Q, R, C}$

Proposition: Product of a Real Number and a Convergent Real Sequence

Let \((a_n)_{n\in\mathbb N}\) be a convergent real sequence with \(\lim_{n\rightarrow\infty} a_n=a\) and let \(x\in\mathbb R\) be a real number. Consider the real sequence \((c_n)_{n\in\mathbb N}\) with \(c_n:=x a_n\). Then \((c_n)_{n\in\mathbb N}\) is also convergent and its limit equals \(\lim_{n\rightarrow\infty} c_n=xa\).

Notes

• This proposition can be expressed in the short form: \[\lim_{n\rightarrow\infty} (x \cdot a_n)=x \cdot \lim_{n\rightarrow\infty} a_n.\]
• The proposition's proof can be transferred also to sequences of other kinds than real numbers, for example, the complex numbers.

Table of Contents

Proofs: 1

Mentioned in:

Proofs: 1 2 3 4 5

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References

Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983