weird sequence

Quote:

Originally Posted by frankdent1

Let a $\epsilon$ (0, 1) be an irrational number. Define a sequence $(x_{n})$ in [0, 1) by $x_{1}$ = 0 and
$x_{n+1}=\left\{\begin{array}{cc}x_{n}+a,&\mbox{ if }<br /> x_{n}+a<1\\x_{n}+a-1, &\mbox{ } otherwise \end{array}\right.$

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

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

If $x_n\to \ell$ then $\ell$ must be equal to either $\ell+a$ or $\ell+a-1$ . You can easily see that neither of these cases is possible.

To find the subsequence, notice that $x_{n+1}$ is the fractional part of $x_n+a$ , and therefore (by induction) $x_{n+1}$ is the fractional part of $na$ . It will be sufficient to show that we can find multiples of a with fractional parts arbitrarily close to 0.

Divide the unit interval into k subintervals of length 1/k. The fractional parts of na are all distinct (otherwise a would be rational), and at least one of the subintervals must contain infinitely many of them. So we can find arbitrarily large m and n (with n>m) for which the fractional parts of ma and na differ by less than 1/k. Then the fractional part of (n-m)a is either less than 1/k, or it is greater than 1-(1/k) in which case the fractional parts of r(n-m)a (for r=1,2,3,...) will move down through the unit interval in steps of less than 1/k, until one of them is less than 1/k.