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!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Convergent Sequences and Functions

**Physics Forums | Science Articles, Homework Help, Discussion**