Example VSIS The vector space of infinite sequences

Set: $\complex{\infty}=\setparts{(c_0,\,c_1,\,c_2,\,c_3,\,\ldots)}{c_i\in\complex{\null},\ i\in\mathbb{N}}$.

Equality: \begin{equation*} (c_0,\,c_1,\,c_2,\,\ldots)=(d_0,\,d_1,\,d_2,\,\ldots)\text{ if and only if }c_i=d_i\text{ for all }i\geq 0 \end{equation*}

Vector Addition: \begin{equation*} (c_0,\,c_1,\,c_2,\,\ldots)+(d_0,\,d_1,\,d_2,\,\ldots)=(c_0+d_0,\,c_1+d_1,\,c_2+d_2,\,\ldots) \end{equation*}

Scalar Multiplication: \begin{equation*} \alpha (c_0,\,c_1,\,c_2,\,c_3,\,\ldots)=(\alpha c_0,\,\alpha c_1,\,\alpha c_2,\,\alpha c_3,\,\ldots) \end{equation*}

This should remind you of the vector space $\complex{m}$, though now our lists of scalars are written horizontally with commas as delimiters and they are allowed to be infinite in length. What does the zero vector look like (Property Z)? Additive inverses (Property AI)? Can you prove the associativity of vector addition (Property AA)?