Does Convergence of d(x_n, x) to 0 Imply x_n Approaches x?

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Homework Statement


show that x_n converges to x if and only d(x_n, x) converges to 0.


Homework Equations


|x_n - x| < ε for all ε>0




The Attempt at a Solution


well d(x_n,x) converges to 0 if d(x_n,x)<ε
i just don't know how to relate that back to |x_n - x|
 
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What is your definition of "converges to"?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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