KöMaL 2018 B4953

Prove that $\ln n+\sqrt{\frac12}+\sqrt{\frac23}+\ldots +\sqrt{\frac{n-1}{n}}<\sqrt2+\sqrt{\frac32}+\sqrt{\frac43}+\ldots +\sqrt{\frac{n}{n-1}}\,$ for all integers $n>1$.