Can Inductive Steps Prove Inequalities in Mathematical Induction?

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Homework Statement



So my question i ...

Homework Equations



... when you're trying to prove some inequality, ...

The Attempt at a Solution



... for example, an < bn, can you do the inductive step by doing legal stuff to both sides of the inequality until you arrive at an+1 < bn+1?
 
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My answer is...
what... is...
legal stuff ? XD ;-)

Do you have a specific example of what you mean ?
 
╔(σ_σ)╝ said:
My answer is...
what... is...
legal stuff ? XD ;-)

Do you have a specific example of what you mean ?

I'm trying to prove that xn < xn+1, where x1 = √2 and xn+1 = √(2 + xn)
 

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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