Let $A=\{\alpha=(a_1,a_2,...):a_n\in\mathbb{C},\lim_{n\rightarrow \infty}a_n\text{ exists }\}$ the set of convergent complex sequences and $||\cdot||_{\infty}$ the sup norm.
How do I prove that $(A,||\cdot||_{\infty})$ is a Banach space?
I know that I have to show that all Cauchy sequences of converging complex sequences converge, but how would I do this?
Hint: Let $(b_n)_{n\in N}$ be a Cauchy sequence of $(A,\|\|_{\infty})$. Write $b_n=(a^m_n)_{m\in N}$ be a Cauchy sequence, show that for every $m$, the sequence $(a^m_n)_{n\in N}$ is a Cauchy sequence. Since complex numbers are complete, the limit $b^m=lim_na^m_n$ exists. Show that $(b_n)_{n\in N}$ converges towards $b=(b^m)_{m\in N}$.