Unbounded Sequence (bn): Convergence of Subsequence (b(kn))

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If (bn) is an unbounded sequence then it has a sequence (b(kn)) such that lim (1/(b(kn)))=0

(where kn is a subsequence of bn )
 
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Hi gankutsuou! Welcome to PF! :smile:

(try using the X2 tag just above the Reply box :wink:)

Do the obvious …

start "If {bn} is unbounded, then for any n … " :smile:​
 

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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