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

# Proposition: Quotient of Convergent Real Sequences

Let $$(a_n)_{n\in\mathbb N}$$ and $$(b_n)_{n\in\mathbb N}$$ be convergent real sequences with the limits $$\lim_{n\rightarrow\infty} a_n=a$$ and $$\lim_{n\rightarrow\infty} b_n=b$$ with $$b\neq 0$$. Then there is a some index $$N\in\mathbb N$$ such that $$b_n\neq 0$$ for all $$n \ge N$$ and the real sequence $$(c_n)_{n\in\mathbb N,~n\ge N}$$ with $$c_n:=a_n/b_n$$ is convergent to the number $$a/b$$.

### Notes

• This proposition can be expressed in the short form: $\lim_{n\rightarrow\infty,~n\ge N} \frac{a_n}{b_n}=\frac{\lim_{n\rightarrow\infty,~n\ge N} a_n}{\lim_{n\rightarrow\infty,~n\ge N} b_n}.$
• The proposition's proof can be transferred also to sequences of other kinds than real numbers, for example, the complex numbers.

Proofs: 1

Proofs: 1 2

Thank you to the contributors under CC BY-SA 4.0!

Github:

### References

#### Bibliography

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