$(12苏大四)$设$fleft( x
ight) in {C^1}left( { - infty , + infty }
ight)$，且[int_{ - infty }^{ + infty } {left[ {f{{left( x
ight)}^2} + {{f'}^2}left( x
ight)}
ight]dx} = 1]
证明：$(1)$$lim limits_{x o egin{array}{*{20}{c}}
infty end{array}} fleft( x
ight) = 0$
$(2)$对任意$x in left( { - infty , + infty } ight)$，有$left| {fleft( x ight)} ight| < frac{{sqrt 2 }}{2}$
$(08华师七)$设$uleft( x
ight)$在$left[ {0, + infty }
ight)$上连续可微，且
[int_0^{ + infty } {left( {{{left| {uleft( x
ight)}
ight|}^2} + {{left| {u'left( x
ight)}
ight|}^2}}
ight)dx} < + infty ]证明：
$(1)$存在$left[ {0, + infty } ight)$上子列$left{ {{x_n}} ight}$，使得${x_n} o infty $，且$uleft( {{x_n}} ight) o 0left( {n o infty } ight)$
$(2)$存在常数$C>0$，使得[mathop {Sup}limits_{x in left[ {0, + infty } ight)} left| {uleft( x ight)} ight| le C{left( {int_0^{ + infty } {{{left| {uleft( x ight)} ight|}^2} + {{left| {u'left( x ight)} ight|}^2}dx} } ight)^{frac{1}{2}}}]