1. ## Monotone Sequences

I could really use some help in one of my questions

Let alpha be a fixed real number such that alpha is strictly less than 1/3. The sequence is defined recusively as follows:

a1 = alpha an+1=1/4(1+an) for n greater or equal to 1

(a) Show that an is less than or equal to 1/3for all n. (Use induction on n.)
(b) Show that (an) is increasing
(c) Deduce that (an) has a limit and determine lim(an) as n->infinity
(d) Show that there exists an nsub0 greater or equal to 1 such that an>0 for all n greater or equal to nsub0

I have done up to (c) but i can't find any notes on part (d). Can anyone help?

Thanks

2. Well, that is basically, the definition of limit isn't it? If $\displaystyle lim a_n= L$, then given any $\displaystyle \epsilon> 0$, there exist N such that if n> N, $\displaystyle |a_n- L|< \epsilon$. Presumably you got a positive number for the limt in (c) (otherewise (d) is not true!). Use the definition of limit with $\displaystyle \epsilon$ equal to that limit.

3. Originally Posted by HallsofIvy
Well, that is basically, the definition of limit isn't it? If $\displaystyle lim a_n= L$, then given any $\displaystyle \epsilon> 0$, there exist N such that if n> N, $\displaystyle |a_n- L|< \epsilon$. Presumably you got a positive number for the limt in (c) (otherewise (d) is not true!). Use the definition of limit with $\displaystyle \color{red}{\epsilon}$ equal to that limit.
I think you meant to say $\displaystyle L$?