MathDB

Suppose

a

and

b

are real numbers such that

0 < a < b < 1

. Let

x= \frac{1}{\sqrt{b}} - \frac{1}{\sqrt{b+a}},\hspace{1cm} y= \frac{1}{b-a} - \frac{1}{b}\hspace{0.5cm}\textrm{and}\hspace{0.5cm} z= \frac{1}{\sqrt{b-a}} - \frac{1}{\sqrt{b}}.

Show that

x

y

z

are always ordered from smallest to largest in the same way, regardless of the choice of

a

and

b

. Find this order among

x

y

z

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