= Bounded sequence with vanishing increments need not converge
{title2=$a_n=\sin\sqrt n$}
The <sequence> is bounded and $|a_{n+1}-a_n|\le\sqrt{n+1}-\sqrt n\to0$ by the <mean value theorem>. Nevertheless, along the integer parts of $(\pi/2+2\pi j)^2$ its terms tend to one, and along the integer parts of $(3\pi/2+2\pi j)^2$ they tend to minus one. The <square root> rounding error is $O(1/j)$. Thus a <bounded sequence> with vanishing successive increments need not be a <convergent sequence>.
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