Question

(3) Suppose that a sequence (an) satifies the condition absolute value of (an+1 - an+2) <1/2 abs value(an - an+1)
Prove that (an) converges.

hint: use the Cauchy criterion, the triangle inequality, and the fact that
 n/2^n approaches 0

Answer 1

i could have sworn i saw this question last night (without the hints)

Answer 2

you have to show it is cauchy
by induction show that
\[|a_{n+2}-a_{n+1}|<\frac{1}{2^n}|a_2-a_1|\]

Answer 3

then you are not done
but i found a nice link that shows if
\[|a_{n+1}-a_n|<\frac{1}{2^n}\] then the sequence is cauchy
let me see if i can find it again
but basically it is multiple triangle inequalities

Answer 4

ok awsome thank you!

Answer 5

http://math.stackexchange.com/questions/182830/proving-that-a-sequence-is-cauchy

Answer 6

ok i will take a look..thank you!

Answer 7

i hope that helps
it is all there, a bit dense, but way more than i could write there
it is repeated triangle inequality, and also summation
pay attentions to the index of summation, it may be a bit confusing (maybe not)

Answer 8

great thank you!