Let

𝑐₀₀ = {(𝑥₁, 𝑥₂, 𝑥₃, … ) ∶ 𝑥_𝑖 ∈ ℝ, 𝑖 ∈ ℕ, 𝑥_𝑖 ≠ 0 only for finitely many indices 𝑖}.

For (𝑥₁, 𝑥₂, 𝑥₃, … ) ∈ 𝑐₀₀, let ‖(𝑥₁, 𝑥₂, 𝑥₃, … )‖_∞ = sup{|𝑥_𝑖 | ∶ 𝑖 ∈ ℕ}.

Define 𝐹, 𝐺 ∶ (𝑐₀₀, ‖⋅‖_∞) → (𝑐₀₀, ‖⋅‖_∞) by

𝐹((𝑥₁, 𝑥₂, … , 𝑥_𝑛, … )) = \(\rm \left((1+x)_{x_1}, (2+\frac{1}{2})x_2,..., (n+\frac{1}{n})x_n,...\right)\)

𝐺((𝑥1, 𝑥2, … , 𝑥𝑛, … )) = \(\rm \left(\frac{x_1}{1+1},\frac{x_2}{2+\frac{1}{2}},..., \frac{x_n}{n+\frac{1}{n}},..\right)\)

for all (𝑥₁, 𝑥₂, … , 𝑥_𝑛, … ) ∈ 𝑐₀₀.

Then