(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let f: ℝ→ℝ be a function such that there exists a constant 0<c<1 for which

|f(x) - f(y)| ≤ c|x - y|

for every x,y[itex]\in[/itex]ℝ.

Prove that there exists a unique a[itex]\in[/itex]ℝ such that f(a) = a.

2. Relevant equations

There's a hint that says: Consider a sequence {x_{n}} defined recurrently by x_{n+1}= f(x_{n}). Prove that it converges and its limit a satisfies f(a) = a.

3. The attempt at a solution

So I did what the hint said, I defined {x_{n}} and proved that it was Cauchy and therefore converges. The problem is after that, I'm not sure if what I did was right, here's what I said:

{x_{n}} converges

Therefore [itex]\exists[/itex]a s.t. lim[itex]_{n→\infty}[/itex]x_{n}= a

Therefore if x_{m}= a, then x_{m+1}= a

Therefore x_{m+1}= a = f(x_{m}) = f(a)

Therefore [itex]\exists[/itex]a s.t. lim[itex]_{n→\infty}[/itex]x_{n}= a and f(a) = a

We know that lim[itex]_{n→\infty}[/itex]x_{n}is unique

Therefore there exists a unique a such that f(a) = a.

I'm not sure if the limit being unique actually implies that a will be the only number such that f(a) = a ?

Also, I'm not sure if, after I state that there exists a limit, I'm allowed to just say that if x_{m}= a, then x_{m+1}= a ? Or is there something I should add in there that I'm not explaining?

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# Homework Help: For f(x), prove that there exists a unique a such that f(a) = a

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