Hello all, I am having trouble with a convergent series problem.(adsbygoogle = window.adsbygoogle || []).push({});

The problem statement:

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 in ℝ. Prove that there exists a unique a in ℝ such that f(a) = a.

There is a provided hint:

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.

My confusion is how to show that the recursive function converges and satisfies the required limit. Should I use a specific function f?

I am fairly certain that we need to show that this sequence (defined in the hint section) is Cauchy. This is what I have so far:

Consider the sequence:

x1 = f(0)

x2 = f(x1)

x3 = f(x2)

...

x(n+1) = f(xn)

Now let us take |xn-xm|:

|xn-xm| = (f(x(n-1)) - f(x(m-1))) < c|x(n-1) - x(m-1)|

However, here I don't know how to pick N sch that for every e>0:

c|x(n-1) - x(m-1)|<e for every n,m>= N (i.e. the definition of a cauchy sequence)

Thanks in advance for all help!

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# Convergent Sequences and Functions

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