= Integral bounds for Euler's constant
{title2=$H_N-\ln(N+1)<\gamma<H_N-\ln N$}
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 $u_N-v_N=\ln(1+1/N)\to0$. 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 $0<1-\ln2<\gamma<1$, since $\ln2<1$.
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