Re: [分析] metric

[THEJOY (最後的演武)]

※ 引述《mathphysics (風的浪漫)》之銘言：
: 各位大大，
: 今天再寫一題簡單分析學中!!
: 有一個題目我要證明它是metric中!
: 要證明三角不等式成立!!
: 可是奇怪我壓不出來!!
: 請板上解答!!
: D(x,y) = d(x,y)/(1+d(x,y)) where d is a metric...
: 感謝版大!!
: 只差三角不等式!

Consider f(t) = t/(1+t), clearly, f is strictly increasing.

Since d is a metric, then d(x,y) ≦ d(x,z) + d(z,y), for any z.

=> D(x,y) = f(d(x,y))

         ≦ f(d(x,z)+d(z,y))

          = (d(x,z)+d(z,y))/(1+d(x,z)+d(z,y))

          = d(x,z)/(1+d(x,z)+d(z,y)) + d(z,y)/(1+d(x,z)+d(z,y))

         ≦ d(x,z)/(1+d(x,z)) + d(z,y)/(1+d(z,y))

          = D(x,z) + D(z,y)

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