TITLE: Euler-Mascheroni constant [strategic proof]
QUESTION [5 upvotes]: I know two proofs about the approximation of Euler-Mascheroni constant $\gamma$ that are very technical.  So I would like to know if someone has a strategic proof to show that $0.5<\gamma< 0.6$.

Let be $\gamma\in \mathbb{R}$ such that
$$\large\gamma= \lim_{n\to +\infty}\left[\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)-\log{(n+1)}\right].$$
Show that $0.5<\gamma< 0.6$

P.S.: In my book, the author use $\log{(n+1)}$ in the limit definition of $\gamma$.

REPLY [2 votes]: Setting $n=1$ and $m=8$ into the following inequality involving harmonic numbers
$$
2H_n-H_{n(n-1)}<\gamma<2H_m-H_{m^2}
$$
gives
$$
0.5<\gamma<0.692
$$