Show that the sequence defined by A1=1 An+1 = 3-1/An is increasing and An<3 for all n?

Question

Deduce that {An} is convergent and find its limit.

Uhhh.. I dont even know how to get started on this one .

Answer 1

A1 = 1

An+1 = 3 - [1/An]

--> An = 3 -[1/An-1]

An is increasing --> An-1 > A1=1
--> 1/An-1 > 0 --> 3 - [1/An-1] <3
or An<3 for all n

limits: An --> 3

Answer 2

To show that it's increasing, show that An+1 - An > 0 for n=1,2,...

Do this by induction. For the general case, note that

An+1 - An = (3 - 1/An) - (3 - 1/An-1) = (1/An-1) - (1/An)

Since by hypothesis An > An-1, we have 1/An < 1/An-1 and so

An+1 - An = (1/An-1) - (1/An) > 0 and so the induction is proved.

You can also show that An < 3 by induction. For the general case, by hypothesis An < 3, so

1/An > 1/3
-1/An < -1/3
3 - 1/An < 3 - 1/3 < 3

But 3 - 1/An = An+1 so An+1 < 3 - 1/3 < 3 and the induction is proved.

So you have a bounded monotone sequence; hence, it converges.

To find the limit, say, A, solve

A = 3 - 1/A for A