Can Contractive Maps and the Sandwich Theorem Prove Sequence Convergence?

[K29]

Homework Statement

This question has been literally troubling me for 2 days.
Question 2(b)
Suppose for some $$c \in R, 0<c<1,$$ we have $$|a_{n+1}-L|<c|a_{n}-L|$$ for $$\forall n \in N$$ (1)

Use the sandwich theorem and the fact that$$\stackrel{lim}{_{n\rightarrow\infty}} c^{n}=0$$ to prove that
$$\stackrel{lim}{_{n \rightarrow \infty}}a_{n}=L$$

Note: I also know and have proven in questions 2(a) that:
$$|a_{n}-L| \leq c^{n} |a_{0} - L| \forall n \in N$$(2)

Note:$$a_{n}$$ is an element of the sequence $$(a_{n})$$

Homework Equations

Definition of a limit:
$$\forall n \in N, \forall \epsilon > 0, \exists K_{ \epsilon} \in R, n \ge K_{\epsilon}$$ such that $$|a_{n}-L|< \epsilon$$

Sandwich theorem:
$$a_{n} \leq b_{n} \leq c_{n}, \stackrel{lim}{_{n \rightarrow \infty}}a_{n} = \stackrel{lim}{_{n \rightarrow \infty}}c_{n}=L \Rightarrow \stackrel{lim}{_{n \rightarrow \infty}}b_{n}=L$$

The Attempt at a Solution

for the sandwich theorem I could have $$b_{n} =c^{n} a_{n}$$. This is less than $$a_{n}$$ since c is between zero and one. However the limit of this is 0(which is also under the assumption that the limit of $$a_{n}$$ exists, which we are trying to prove. So starting off with sandwich theorem basics doesn't really help me because I also can't think of any $$c_{n} \geq a_{n}$$.

The next thing I tried is considering the statements in the problem.
$$|a_{n}-L| \leq c^{n} |a_{0} - L|$$ could have been useful. But it doesn't imply anything about the sequence and its limits. I.E. I can't assume that this implies that $$a_{n} \leq c^{n}a_{0}$$

I can't really apply the definition of a limit to anything. By making the assumption that any of the equations above are less than some$$\epsilon$$ almost automatically gives the result that the limit for $$a_{n}$$ is L. i.e assuming the right hand side of equation (2) is less than some epsilon for $$\forall n>K$$means$$|a_{n}-L|<\epsilon$$ holds true as well.

Well it would almost hold true. But without knowing the details of $$a_{n}$$ I caan't prove it to be true. I.E I can't construct a $$K_{\epsilon}$$ unless I know more about $$a_{n}$$

PLEASE help on this one. I've been struggling for ages. Thanks

 

[Stephen Tashi]

Just a thought - Do your materials have a theorem that says
If $$lim_{n \rightarrow \infty} | a_n - L | = 0$$ then $$lim_{n \rightarrow \infty} a_n= L$$ ?

If so, you could consider defining 3 sequences by
$$\{b_n\} = 0$$
$$\{|A_n - L|\}$$
$$\{c^n | A_0 - L |\}$$

 

[K29]

My materials don't have it, but I easily proved it and the proof is so short this must be right. Thanks!