假设$forall x in left( {a, + infty } ight),f'left( x ight) e 0$,则由$f{Darboux定理}$知,$f'left( x ight)$恒正或恒负
不妨设$f'left( x
ight)$恒正,则$fleft( x
ight)$严格单调增加,于是
[fleft( x
ight) > fleft( {a + 2}
ight) > fleft( {a + 1}
ight) > fleft( y
ight),forall x > a + 2,y in left( {a,a + 1}
ight)]
从而可知[mathop {lim }limits_{x o egin{array}{*{20}{c}}
{ + infty }
end{array}} fleft( x
ight) ge fleft( {a + 2}
ight) > fleft( {a + 1}
ight) ge mathop {lim }limits_{x o egin{array}{*{20}{c}}
{{a^ + }}
end{array}} fleft( x
ight)]
这与题设条件矛盾