Consider the set $V$ consisting of all infinite sequences $a = (a_0, a_1, \ldots)$ where each $a_i \in \mathbb{R}$ and $a_0 \neq 0$. How can we demonstrate that the operation $(a + b) + c = a + (b + c)$ is associative for all $a, b, c \in V$?
$a + b = (a_0b_0, a_0b_1 + a_1b_0, \ldots)$ and it is equal to: $(a + b)_j = \sum_{i=0}^{j}a_ib_{j−i}$
I attempted to prove this by expanding both sides into sums, but I suspect this method might not be the most efficient or revealing. Is there a more innovative or insightful approach to addressing this problem? I would greatly appreciate any guidance or alternative methods for proving this property.
Thank you!
Consider each sequence as the coefficients of a formal power series with non-zero constant terms. The "sum" operation you're working is simply multiplication of two such series, so associativity of the sum follows from associativity of ordinary multiplication of power series.