MathDB

Let

\{a_n\}

be a bounded real sequence. (a) Prove that if X is a positive-measure subset of

\mathbb R

, then for almost all

x\in X

, there exist a subsequence

\{y_n\}

of X such that

\sum_{n=1}^\infty (n(y_n-x)-a_n)=1

(b) construct an unbounded sequence

\{a_n\}

for which the above equation is also true.

6