Question:
Let $(X, d)$ be a metric space, let $f : X → X$ be a contraction, and
let $a_o \in X$. Let $a_1 = f(a_o)$ and $a_{n+1} = f(a_n)$ for $n \geq
1$. Prove that $(a_n)$ is a Cauchy sequence.
Attempt:
Given contraction constant $k$$$ 0 \leq k < 1$$
First note that by definition of contraction $$kd(a_2,a_1) \geq d(f(a_2), f(a_1)) = d(a_3, a_2)$$
Then
$$k^2d(a_2,a_1) \geq kd(a_3,a_2) \geq d(f(a_3), f(a_2)) = d(a_4, a_3)$$
$$k^3d(a_2,a_1) \geq k^2d(a_3,a_2) \geq kd(a_4, a_3) \geq d(a_5,a_4)$$
Therefore, $$k^n d(a_2,a_1) \geq d(a_{n+2}, a_{n+1})$$
Then to show that $(a_n)$ is Cauchy, we must show that $\forall \epsilon > 0, \exists N$ such that $\forall n, m \geq N, d(a_n, a_m) < \epsilon$
Then by triangle inequality:
$$d(a_n, a_m) \leq d(a_{n+1},a_{n}) + d(a_{n+2}, a_{n+1}) + \ldots d(a_{m-1}, a_{m})$$
Let $$k^n d(a_2,a_1) = k^n r \geq d(a_{n+2}, a_{n+1}), r = d(a_2, a_1)$$
How do I use the above to provide a bound on $d(a_n, a_m)$?
Note: some stuff maybe messed up in the above derivation, fixing ….
Best Answer
Like you say above $$d(a_n,a_m) \leq \sum_{j=n}^{m-1} d(a_j,a_{j+1}) \leq \sum_{j=n}^{m-1} d(a_1,a_2)\cdot k^{j-1} \leq d(a_1,a_2) \sum_{j=n}^{\infty} k^{j-1} = d(a_1,a_2) \frac{k^{n-1}}{1-k}$$
Thus given $\epsilon>0$, choose $N$ such that $d(a_1,a_2)\frac{k^{N-1}}{1-k}< \epsilon$. Then clearly, $m> n \geq N$ imply that $d(a_n,a_m) \leq d(a_1,a_2) \frac{k^{n-1}}{1-k}<\epsilon$.