Let \((a_n)_{n\in\mathbb N}\) and \((b_n)_{n\in\mathbb N}\) be complex sequences convergent to the limits \(\lim_{n\rightarrow\infty} a_n=a\) and \(\lim_{n\rightarrow\infty} b_n=b\). Consider the complex sequence \((c_n)_{n\in\mathbb N}\) with \(c_n:=a_n \cdot b_n\). Then \((c_n)_{n\in\mathbb N}\) is also convergent and its limit equals \(\lim_{n\rightarrow\infty} c_n=a \cdot b\).
This proposition can be expressed in the short form:
\[\lim_{n\rightarrow\infty} (a_n \cdot b_n)=\lim_{n\rightarrow\infty} a_n \cdot \lim_{n\rightarrow\infty} b_n.\]
Proofs: 1
