已知数列${a_n}$满足:$a_n>0,a_{n+1}+dfrac{1}{a_n}<2,nin N^*$.
求证:
(1)$a_{n+2}<a_{n+1}<2 (nin N^*)$
(2)$a_n>1 (nin N^*)$
第二题:分析:由题意${a_n}$单调递减又有下界,故有极限,记$limlimits_{nlongrightarrow +infty}a_n=x$
则由$a_{n+1}+dfrac{1}{a_n}<2$两边取极限得$x+dfrac{1}{x}le2$,又由于$x+dfrac{1}{x}ge2$故$limlimits_{nlongrightarrow +infty}a_n=1$
由单调递减得$a_n>1$
注:也可以用反证法，提示:关键递推式$dfrac{1}{a_{n+1}-1}>1+dfrac{1}{a_n-1}$