Proof verification. $\{x_n\}$ is a sequence such that $|x_{n+1} - x_n| \le C\alpha^n$ for $\alpha\in (0, 1), n\in\Bbb N$. Prove $x_n$ converges.

Question

Let $\{x_n\}, n\in \Bbb N$ denote a sequence such that: $$ \begin{cases} |x_{n+1} - x_n| \le C\alpha^n \\ 0 < \alpha < 1 \end{cases} $$ Prove $\{x_n\}$ converges.

Given the fact $|x_{n+1} - x_n| \le C\alpha^n$ consider the following inequalities: $$ |x_{n+1} - x_n| \le C\alpha^n \\ |x_{n+2} - x_{n+1}| \le C\alpha^{n+1} \\ |x_{n+3} - x_{n+2}| \le C\alpha^{n+2} \\ \dots \\ |x_{n+p+1} - x_{n+p}| \le C\alpha^{n+p} \\ $$

Consider the sum of the inequalities: $$ |x_{n+1} - x_{n}| + |x_{n+2} - x_{n+1}| + |x_{n+3} - x_{n+2}| + \cdots + |x_{n+p+1} - x_{n+p}| \\ \le \sum_{k=0}^p C\alpha^{n+k} = C \sum_{k=0}^p \alpha^{n+k} \tag1 $$

By geometric sum: $$ C \sum_{k=0}^p \alpha^{n+k} = C \cdot \frac{\alpha^n(1-\alpha^{p + 1})}{1-\alpha} \le C \cdot \frac{\alpha^n}{1-\alpha} $$

Lets fix some $\epsilon > 0$, and $N\in \Bbb N$ such that: $$ C\cdot \frac{\alpha^{N}}{1-\alpha} < \epsilon $$

Rewrite $\alpha$ as: $$ \alpha = \frac{1}{1+r},\ r \in \Bbb R_{>0} $$

Thus: $$ C\cdot \frac{1}{(1-\alpha)(1+r)^N} < \epsilon \\ (1+r)^N > {C\over (1-\alpha)\epsilon} \\ N > \log_{1+r} {C\over (1-\alpha)\epsilon} $$

Returning to $(1)$ we have by triangular inequality: $$ |x_{n}- x_{n+1} + x_{n+1} - x_{n+2} + \cdots + x_{n+p} - x_{n+p+1}| \\ \le |x_{n+1} - x_{n}| + |x_{n+2} - x_{n+1}| + \cdots + |x_{n+p+1} - x_{n+p}| $$

Since the values are telescoping we obtain: $$ |x_{n} - x_{n+p+1}| < C \cdot \frac{\alpha^n}{1-\alpha} < \epsilon $$

Now if we choose: $$ n > N > \log_{1+r} {C\over (1-\alpha)\epsilon} $$ we obtain a regular definition of the Cauchy criteria, which means $\{x_n\}$ is a convergent sequence.

Could you please verify the reasoning above? Thank you!

Answer 1

You have the right idea but it could be briefer: Let $A(n)=\sup_{m\in \Bbb N}|x_n-x_{n+m}|.$ The Cauchy Criterion for convergence of $(x_n)_{n\in \Bbb N}$ is $\lim_{n\to \infty}A(n)=0.$

Since $0<\alpha<1$ we have $\sup_{m\in \Bbb N}(1-\alpha^m)=1,$ so $$A(n)=\sup_{m\in \Bbb N}|x_n-x_{n+m}|\leq$$ $$\leq \sup_{m\in \Bbb N}C\alpha^n\cdot\frac {1-\alpha^m}{1-\alpha}=$$ $$=C\alpha^n\cdot \frac {1}{1-\alpha}.$$ This last expression goes to $0$ as $n\to \infty$ because $0<\alpha <1$.

Answer 2

Seems fine.

We have $C \ge 0$.

In the event that $C=0$, we have a constant sequence and hence it converges.

Answer 3

This is a comment rather than an answer.

You don't have to use logs.

By Bernoulli's inequality, $(1+r)^N \ge 1+rN \gt rN$ so that $C \frac{1}{(1-\alpha)(1+r)^N} \lt C\frac{1}{(1-\alpha)rN} $.

Therefore, if $\epsilon \gt C \frac{1}{(1-\alpha)rN} $, or $N > \frac{C}{(1-\alpha)r\epsilon} $, then $C \frac{1}{(1-\alpha)(1+r)^N} < \epsilon $.

Of course this isn't as good as the equation using logs, but it is completely elementary.