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