November 21st, 2011, 05:15 PM  #1 
Member Joined: Oct 2011 Posts: 35 Thanks: 0  Intermediate Values
Suppose that f is continuous on [0,1], and maps [0,1] into [0,1] Take any point c in [0,1] and define a sequence {Xn} inductively by X1= c, Xn+1=f(Xn). Suppose that {Xn} converges to the point p. Prove that p is a fixed point of f(x) I have no idea how this works.. smart people please HELP! 
November 21st, 2011, 05:44 PM  #2 
Member Joined: Dec 2010 From: Miami, FL Posts: 96 Thanks: 0  Re: Intermediate Values
Since is a continuous function and , the following folds: 
November 21st, 2011, 06:08 PM  #3 
Member Joined: Oct 2011 Posts: 35 Thanks: 0  Re: Intermediate Values
any idea on how to prove p is a fixed point of f(x)?

November 21st, 2011, 06:15 PM  #4 
Member Joined: Dec 2010 From: Miami, FL Posts: 96 Thanks: 0  Re: Intermediate Values
Consider and apply what I mentioned above.

November 21st, 2011, 06:28 PM  #5 
Member Joined: Oct 2011 Posts: 35 Thanks: 0  Re: Intermediate Values
i'm slowly trying to piece this together.. so if n=1 then Xn+1 = X2 = f(X1) if n=2 then Xn+1 = X3 = f(X2) = f(f(X1)) if n=3 then Xn+1 = X4 = f(X3) = f(f(f(X1))) and so on... maybe? 
November 21st, 2011, 06:43 PM  #6 
Member Joined: Dec 2010 From: Miami, FL Posts: 96 Thanks: 0  Re: Intermediate Values
We want to show . We know that , so that , and . We just have to piece it all together. 
November 21st, 2011, 07:00 PM  #7 
Member Joined: Oct 2011 Posts: 35 Thanks: 0  Re: Intermediate Values
i'm making a mess.. i have: Prove p is a fixed point. I want to show that f(p)=p I know that Xn>p so f(p)=f(lim n>oo Xn) and Xn+1=f(Xn) so f(lim n>oo Xn+1) = f(lim n>oo f(Xn))= f(f(lim n>oo Xn))= f(f(p)) 
November 21st, 2011, 07:14 PM  #8 
Member Joined: Dec 2010 From: Miami, FL Posts: 96 Thanks: 0  Re: Intermediate Values
You are very close. Apply what I mentioned to you earlier before you apply 
November 21st, 2011, 07:57 PM  #9 
Member Joined: Oct 2011 Posts: 35 Thanks: 0  Re: Intermediate Values
i'm really not close.. i have: applying Xn+1 = f(Xn) so f(Xn+1)=f(f(Xn)) taking the limit of both we have lim n>oo f(Xn+1) = f(f(p)) and lim n>oo f(f(Xn))= f(f(p)) i'm not sure how you want me to apply Xn+1 = f(Xn) 
November 21st, 2011, 08:34 PM  #10  
Member Joined: Dec 2010 From: Miami, FL Posts: 96 Thanks: 0  Re: Intermediate Values Quote:
 

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