Proof explaination - $\sum_{i=1}^{n} \frac{1}{i}$ is not an integer for $n>1$

Question

I was reading a proof to the following fact: for $n>1$, $\sum_{i=1}^{n} \frac{1}{i} \notin \mathbb{Z}$.
The proof is as follows: Denote for prime $p$ by $v_p(a)$ the p-adic valuation of $a$.
Write the sum as $\sum_{i=1}^{n}\frac{\frac{n!}{i}}{n!}$. We will show that $2$ divides the denominator more times that $2$ divides the numerator. We consider $v_2(\sum_{i=1}^{n} \frac{n!}{i})$. By the theorem $v_p(a) > v_p(b) \rightarrow v_p(a+b)=v_p(b)$ we get $v_2(\frac{n!}{2i-1}+\frac{n!}{2i})=v_2(\frac{n!}{2i})$. We then get $v_2(\frac{n!}{4i-1}+\frac{n!}{4i})=v_2(\frac{n!}{4i})$ and repeating to sum up the factorial in this way we arrive at $v_2(\sum_{i=1}^{n} \frac{n!}{i})=v_2(\frac{n!}{2^{[log_2(n)]}})$
However for $\sum_{i=1}^{n} \frac{1}{i}$ to be an integer we need $v_2(\sum_{i=1}^{n} \frac{n!}{i}) \geq v_2(n!)$ which is the same as $v_2(\frac{n!}{2^{[log_2(n)]}}) \geq v_2(n!)$ which is the same as $0 \geq v_2(n!)$ which is not true for $n>1$.

I didn't understand this proof. I followed until he said "and repeating to sum up the factorial in this way we arrive at...". What is meant there? Can someone explain in more detail what's happening?

Answer 1

Here is an easier way to understand the proof:

Let $k$ be so that $2^k \leq n <2^{k+1}$.

Write $$\sum_{i=1}^{n} \frac{1}{i}= \left(\sum_{1 \leq i \leq n ; i \neq 2^k} \frac{1}{i} \right)+\frac{1}{2^k}$$

Now, bring at the same denominator $$\left(\sum_{1 \leq i \leq n ; i \neq 2^k} \frac{1}{i} \right)=\frac{A}{B}$$ where $A,B$ is reduced.

Show that $2^k \nmid B$. Deduce from here that

$$\frac{A}{B}+\frac{1}{2^k}=\frac{C}{D}$$ $2^k |D$ but $2^k \nmid C$.