证明：令$d = mathop {inf}limits_{y in M} left| {x - y} ight|$，由下确界的定义知，存在${x_n} in M$，使得[mathop {lim }limits_{n o infty } left| {{x_n} - x} ight| = d]
下面我们证明$left{ {{x_n}} ight}$是基本列.由平行四边形公式知[{left| {{x_m} - x} ight|^2} + {left| {{x_n} - x} ight|^2} = 2left( {{{left| {frac{{{x_m} + {x_n}}}{2} - x} ight|}^2} + {{left| {frac{{{x_m} - {x_n}}}{2}} ight|}^2}} ight)]由于$frac{{{x_m} + {x_n}}}{2} in M$，则$left| {frac{{{x_m} + {x_n}}}{2} - x} ight| ge d$，从而可知[0 le frac{1}{2}{left| {{x_m} - {x_n}} ight|^2} le {left| {{x_m} - x} ight|^2} + {left| {{x_n} - x} ight|^2} - 2{d^2}]令$n,m o infty $，则$left| {{x_m} - {x_n}} ight| o 0$，所以$left{ {{x_n}} ight}$是基本列
又由于$M$为$f{Hilbert}$空间$X$的闭子空间，则存在${x_0} in M$，使得${x_n} o {x_0}$，此时[left| {x - {x_0}} ight| = mathop {lim }limits_{n o infty } left| {x - {x_n}} ight| = d]
下面我们证明$x - {x_0} ot M$.由线性子空间的定义知，对任意的$z in M,z e 0$，以及$lambda in K$，有${x_0} + lambda z in M$，于是[left| {x - left( {{x_0} + lambda z} ight)} ight| ge d]所以有[{left| {left( {x - {x_0}} ight) - lambda z} ight|^2} = {left| {x - {x_0}} ight|^2} - 2{mathop{ m Re} olimits} left( {overline lambda left( {x - {x_0}} ight),z} ight) + {left| lambda ight|^2}{left| z ight|^2} ge {d^2}]令$lambda = frac{{left( {x - {x_0},z} ight)}}{{{{left| z ight|}^2}}}$，则有
[{left| {x - {x_0}} ight|^2} - 2frac{{{{left| {left( {x - {x_0},z} ight)} ight|}^2}}}{{{{left| z ight|}^2}}} + frac{{{{left| {left( {x - {x_0},z} ight)} ight|}^2}}}{{{{left| z ight|}^2}}} = {left| {x - {x_0}} ight|^2} - frac{{{{left| {left( {x - {x_0},z} ight)} ight|}^2}}}{{{{left| z ight|}^2}}} ge {d^2}]由$left| {x - {x_0}} ight| = d$可知，$left( {x - {x_0},z} ight) = 0$，即$x - {x_0} ot M$
下面我们证明唯一性.假设还存在${x_0}^prime in M$,${x_1}^prime in {M^ ot }$，使得[x = {x_0}^prime + {x_1}^prime ]则${x_0} - {x_0}^prime in M,{x_1} - {x_1}^prime in {M^ ot }$，从而由[{x_1} - {x_1}^prime = left( {x - {x_0}} ight) - left( {x - {x_0}^prime } ight) = {x_0}^prime - {x_0} in M]可知${x_1}^prime = {x_1},{x_0}^prime = {x_0}$