Throughout the problem, we denote for every integer $n \geqslant 1$, $H _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k } = 1 + \frac { 1 } { 2 } + \cdots + \frac { 1 } { n }$.
Show that there exists a real constant $A$ such that $H _ { n } \underset { + \infty } { = } \ln n + A + o ( 1 )$. Deduce that $H _ { n } \sim \ln n$.