Show that the sequence $H_n - \lfloor H_n \rfloor$ is not convergent

Question

Let $x_n = 1 + \frac{1}{2} + \dots + \frac{1}{n} - \left\lfloor 1 + \frac{1}{2} +\dots+\frac{1}{n}\right\rfloor \ \forall n \in \mathbb{N} $ be a sequence . Prove that it is not convergent?

$\lfloor x\rfloor$ means floor.

I have absolutely no ideea how to prove that is not convergent ??

Answer 1

Using the standard notations $\{x\}=x-\lfloor x\rfloor$ for the fractional part and $H_n=\sum_{k=1}^{n}\frac{1}{n}$ for harmonic numbers, we are asked to prove that the sequence of fractional parts of harmonic numbers is not convergent. That can be proved, for instance, by showing that every element $\alpha\in(0,1)$ is an accumulation point for such a sequence. $\{H_n\}_{n\geq 1}$ is a slowly increasing sequence since $H_n=\log(n)+\gamma+O\left(\frac{1}{n}\right)$. If we choose $n$ as the closest integer to $e^{M-\gamma+\alpha}$, with $M$ being a huge natural number, then $\{H_n\}$ is arbitrarily close to $\alpha$ and we are done.

Answer 2

Jack's proof applies to any sequence defined by $x_n = s_n -\lfloor s_n \rfloor$ where $s_n =\sum_{k=1}^n t_k$ with $t_n >0, t_{n+1}<t_n, t_n \to 0$. and $s_n \to \infty$.

Then $x_n$ will be dense in $[0,1]$ as can be seen by noting that, for any integer $m>0$, once $t_n<1/m$, every value in $[0,1]$ will be within $1/m$ of a $t_{n+k}$.