Let $\mathbb{R}_{\ge 0}^{*} = [0, \infty]$ be the extended non-negative reals and multiplication be defined so that $0 \cdot \infty = \infty \cdot 0 = 0$. Does the following identity hold for any $k, a_i \in \mathbb{R}^{*}_{\ge 0}$?$$ \sum_{i = 1}^\infty ka_i = k\sum_{i = 1}^\infty a_i.$$My guess is yes because there is only one sequence that sums to $0$ (the constant sequence $(0, 0, 0, ...)$) and it yields $0$ on both sides.
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