Let a $\displaystyle \epsilon$ (0, 1) be an irrational number. Define a sequence $\displaystyle (x_{n})$ in [0, 1) by $\displaystyle x_{1}$ = 0 and

$\displaystyle x_{n+1}=\left\{\begin{array}{cc}x_{n}+a,&\mbox{ if }

x_{n}+a<1\\x_{n}+a-1, &\mbox{ } otherwise \end{array}\right.$

Show that $\displaystyle x_{n}$ does not converge.

Show that there is some subsequence of $\displaystyle x_{n}$ which converges to 0.