Let \((a_n)_{n\in\mathbb N}\) be a convergent complex sequence with \(\lim_{n\rightarrow\infty} a_n=a\) and let \(x\in\mathbb R\) be a complex number. Consider the complex 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 \(\lim_{n\rightarrow\infty} c_n=xa\).
This proposition can be expressed in the short form:
\[\lim_{n\rightarrow\infty} (x \cdot a_n)=x \cdot \lim_{n\rightarrow\infty} a_n.\]
Proofs: 1
Proofs: 1
