SCHWEITZER 1996 Problem 6

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.