已知数列 $\left\{\mathrm{a}_n\right\}$ 中, $\mathrm{a}_1=1$, 若 $a_{n+1}=\frac{(n+1) a_n}{n+1+a_n}$, 则下列结论中正确的是
A. $\frac{1}{a_{n+1}}-\frac{1}{a_n} \geq \frac{1}{2}$
B. $\frac{1}{a_{n+2}}-\frac{1}{a_n} < \frac{2}{\sqrt{(n+2)(n+1)}}$
C. $\frac{1}{a_{2 n}}-\frac{1}{a_n} \geq \frac{1}{2}$
D. $a_n \cdot \ln (n+1)>1$