EDB — 0F8

Exercises

1. [0F8]A sequence is given \((a_ n)_{n∈ {\mathbb {N}}}\) of positive real numbers such that \(\lim _{n→∞} a_ n=0\) and \(∑_{n=0}^∞ a_ n=∞\): prove that for every \(l ∈ {\mathbb {R}}\) there is a sequence \((ε_ n )_{n∈{\mathbb {N}}}\) with \(ε_ n ∈\{ 1,-1\} \) for each n, such that

   \[ ∑_{n=0}^∞ (ε_ n a_ n)=l\quad . \]

   If instead \(∑_{n=0}^∞ a_ n=S{\lt}∞\), what can be said about the set \(E\) of the sums \(∑_{n=0}^∞ (ε_ n a_ n)=l\), for all possible choices of \((ε_ n )_{n∈{\mathbb {N}}}\) with \(ε_ n ∈\{ 1,-1\} \) for every n?

   • Analyze cases where \(a_ n=2^{-n}\) or \(a_ n=3^{-n}\)

   • Show that \(E\) is always closed.

   • Under what assumptions do you have that \(E=[-S,S]\)?

   Hint. Let \(\tilde E\) be the set of sums \(∑_ n (ε_ n a_ n)=l\), to vary by \((ε_ n )_{n∈{\mathbb {N}}}\) with \(ε_ n ∈ \{ 0,1\} \) for each n; note that \(\tilde E=\{ (S+x)/2 : x∈ E\} \).