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# Sequence - Limit watch

1. So I am doing like that

We have proved -1 < xn <= -1/3
And we have

Let f(x) = x4 - 4x² - 4
=> f'(x) = 4x³ - 8x = 4x(x - √2)(x+√2)
For x E (1, -1/3], then f'(x) > 0
=> f(x) is increasing for x in (1, -1/3]
i.e x1 < x2 and x1, x2 E (1, -1/3] => f(x1) < f(x2)

Prove x2n is increasing by induction

Similarly, we can prove x2n+1 is decreasing sequence.

Then now we have to prove 2 subsequences have the same limit

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Updated: November 23, 2005
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