# Sum of bijective functions

Let $f,g_{1}, g_{2},\ldots,g_{k} \in \mathbb{Q}^{\mathbb{N}}$.

$f$ is a sum of $g_{1},g_{2},\ldots,g_{k}$ if for every natural number $n$

$$f(n) = g_{1}(n) +g_{2}(n) + \cdots + g_{k}(n)$$

a) Prove that for every $f \in \mathbb{Q}^{\mathbb{N}}$, $\exists$ three bijections $g_{1}, g_{2} , g_{3} \in \mathbb{Q}^{\mathbb{N}}$, such that $f$ is a sum of $g_{1}, g_{2},g_{3}$.

b) Give example for $f \in \mathbb{Q}^{\mathbb{N}}$, that is not a sum of two bijections $g_{1}, g_{2} \in \mathbb{Q}^{\mathbb{N}}$.

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A quick Google search revealed this: mathoverflow.net/questions/56430/sum-of-three-bijections – Beni Bogosel Feb 24 '12 at 22:57
@Beni : this could be an answer. +1 for me! – Patrick Da Silva Feb 24 '12 at 23:04
This must be an annual thing. The answers were posted to the MO question exactly one year ago today. – Gerry Myerson Feb 25 '12 at 3:29
@Gerry: Interestingly, the MO question uses different variable names (and has $\mathbb Z$ where $\mathbb N$ seems to have been intended), yet the two questions have the same inappropriate indefinite article in "a sum". – joriki Feb 25 '12 at 6:35