29595959

证明：(1)因为$X_0$为$X$的真子空间，于是存在${x_1} in Xackslash {X_0}$，记$$d = mathop {inf }limits_{x in {X_0}} left| {x - {x_1}} ight|$$

(2)因为$X_0$是闭的，故$d>0$，否则存在${x_n} in {X_0}$，且$left| {{x_n} - {x_1}} ight| o 0$，再由$X_0$是闭的推出${x_1} in {X_0}$矛盾

(3)不妨设$varepsilon  < 1$，则有$frac{d}{{1 - varepsilon }} > d$，由下确界的定义知，存在${x_2} in {X_0}$，使得[left| {{x_2} - {x_1}} ight| < frac{d}{{1 - varepsilon }}]

(4)令${x_0} = frac{{{x_1} - {x_2}}}{{left| {{x_1} - {x_2}} ight|}}$，则$left| {{x_0}} ight| = 1$，对于任何$x in {X_0}$，注意到${x_2} in {X_0}$，我们有

egin{align*}
left| {x - {x_0}} ight|& = left| {x - frac{{{x_1} - {x_2}}}{{left| {{x_1} - {x_2}} ight|}}} ight| = frac{1}{{left| {{x_1} - {x_2}} ight|}}left| {left( {left| {{x_1} - {x_2}} ight|x + {x_2}} ight) - {x_1}} ight|\&
ge frac{1}{{left| {{x_1} - {x_2}} ight|}} cdot d > 1 - varepsilon
end{align*}