EDB — 0F5

Exercises

1. [0F5]We indicate with \({\mathcal P}_{\mathfrak f}({\mathbb {N}})\) the set of subsets \(B⊆ {\mathbb {N}}\) which are finite sets. This is said the set of finite parts.

   We abbreviate \({\mathcal P}={\mathcal P}_{\mathfrak f}({\mathbb {N}})\) in the following.

   Given a sequence \((a_ n)_ n\) of real numbers and a \(B∈{\mathcal P}\) we indicate with \(s(B)=∑_{n∈ B} a_ n\) the finite sum with indices in \(B\).

   Suppose the series \(∑_{n=0}^∞ a_ n\) converge but not converge at all. Then:

   • \(\{ s(F ) : F ∈{\mathcal P}\} \) it is dense in \({\mathbb {R}}\).

   • There is a reordering \(σ\) of \({\mathbb {N}}\), that is, a bijective function \(σ:{\mathbb {N}}→{\mathbb {N}}\), such that all partial sums \(∑_{n=0}^ N a_{σ(n)}\) (at the variation of \(N\)) is dense in \({\mathbb {R}}\).

   [ [0F6]↺↻]