Integral bounds for Euler's constant ($H_N-\ln (N+1)<\gamma <H_N-\ln N$)

For the harmonic number $H_N$, the upper sequence $u_N=H_N-\ln N$ decreases and the lower sequence $v_N=H_N-\ln (N+1)$ increases. The inequalities $\ln (1+1/N)>1/(N+1)$ and $\ln (1+1/(N+1))<1/(N+1)$ follow by integrating the decreasing function $1/x$ over a unit interval. They give the two monotonicities, while $u_N-v_N=\ln (1+1/N)\to 0$. The monotone convergence theorem for sequences therefore gives the common limit Euler's constant $\gamma$. Strict monotonicity makes each finite-index bound strict. At $N=1$, this gives $0<1-\ln 2<\gamma <1$, since $\ln 2<1$.