Let \((a_n)_{n\in\mathbb N}\) and \((b_n)_{n\in\mathbb N}\) be convergent complex 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 complex sequence \((c_n)_{n\in\mathbb N,~n\ge N}\) with \(c_n:=a_n/b_n\) is convergent to the number \(a/b\).
This proposition can be expressed in the short form:
\[\lim_{n\rightarrow\infty,~n\ge N} \frac{a_n}{b_n}=\frac{\lim_{n\rightarrow\infty,~n\ge N} a_n}{\lim_{n\rightarrow\infty,~n\ge N} b_n}.\]
Proofs: 1
Proofs: 1
