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Homework Statement
Suppose that x0[tex]\in[/tex] (-1,0) and xn=[tex]\sqrt{xn-1+1}[/tex] -1 for n[tex]\in[/tex]N. Prove that xn[tex]\uparrow[/tex]0 as n[tex]\rightarrow[/tex][tex]\infty[/tex].
the formatting is odd so read outloud, the function for x sub n is "x sub n equals sqrt(x sub (n-1) + 1) -1. If it is still unclear, ask and I can post an image of it typed in maple or something.
Homework Equations
There are several (many) theorems which could be applied but all of them are well known and prety trivial/obvious.
The Attempt at a Solution
Basically I'm having trouble with the "second" part of this. I proved by induction that the sequence is monotonically increasing but I am unable to prove that the limit is 0. I've tried directly proving it using the definition of a limit of a sequence (given [tex]\epsilon[/tex]>0, there is an n...) but am unable to do so because of the fact that this is a recursive function and not a function of n. I've also tried proving this logically by trying to show that 0=sup{xn} but again the fact that this is a recursive function inhibits this. One more note: this is problem number 1 in my text and usually (always) the problems get harder as you go along which makes me think I am missing something very trivial here. Any ideas?
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