Let $a(n)=\sum_{r=1}^{n}(-1)^{r-1}\frac{1}{r}$. Prove that $a(2n) \neq 1$ for any value $n$. Let $$a(n)=\sum_{r=1}^{n}(-1)^{r-1}\frac{1}{r}.$$
I experienced this function while doing a problem, I could do the problem, but I got stuck at a point where I had to prove that $a(2n)<1$ for all $n$. I proved that $a(2n) \le 1$ for all $n$ so in order to prove what the question demands, I need to prove that $a(2n) \neq 1$ for any value $n$. Nothing striked my mind as of now how to prove it.
So can someone help me proving that $a(2n) \neq 1$ for any value  $n$?

$\frac{1}{n}$ is strictly decreasing, so $a(2n)$ is strictly increasing. $$\lim_{n\to\infty}a(2n)=\ln 2,$$ so
$$0\leq a(2n)<\ln 2< 1$$
