Proving a Sequence is Convergent

[bobcat817]

Homework Statement

Let {a_{n}}^{\infty}_{n=1} be a sequence of real numbers that satisfies


|a_{n+1} - a_{n}| \leq \frac{1}{2}|a_{n} - a_{n-1}|

for all n\geq2

Homework Equations

The Attempt at a Solution

So, I know that it suffices to show that the sequence is Cauchy to prove this. And I can show that

|a_{m} - a_{n}| \leq \sum^{m-1}_{n}\frac{1}{2}^{k-1} * |a_{2} - a_{1}|

And that is about where I get lost. I'm very unclear on how to define the N in relation to the \epsilon.

 

[statdad]

I will take your comment that you've shown

<br /> |a_m - a_n | \le |a_2 - a_1| \sum_{k=n}^m \left(\frac 1 2 \right)^k<br />

as correct - make sure it's correct.

Suppose \varepsilon &gt; 0 is given. Try taking N so large that n, m larger than N give

<br /> \sum_{k=n}^m \left(\frac 1 2 \right)^k &lt; \frac{\varepsilon}{2|x_2 - x_1|}<br />

This should work (why the 2 in my denom - to make sure the sequence terms differ by less than \varepsilon - it's a holdover from my student days, and I'm too old to change.)