Show sequence is a cauchy sequence.

Question

If $C_n$ is a bounded sequence and $D_n$ is defined by $D_n = \sum_{k=1}^n C_kQ^{-k}$. Then show if $Q>1$ then $(D_n)$ is a Cauchy sequence.

Answer 1

Note that $C_n$ is bounded, say $|C_n| \le C$ for all $n$. You want to show, given $\epsilon>0$, that there exists $N$ such that $$\forall n,m \ge N, \ |D_n - D_m| < \epsilon.$$

So manipulate the expression. For $n>m$ (without loss of generality)

$$|D_n - D_m| = | \sum_{k=m+1}^n C_n Q^{-k} |.$$

There is a little more work that I haven't included as it's better for you do do than me tell you. All you need is bound the final term, then show the bound goes to $0$. Note that since $Q>1$, $C \sum_{k=0}^\infty Q^{-k} < \infty$.

Hopefully this hint should get you there!