Homework Help: About Convergence of a particular sequence

1. Sep 19, 2010

evalover1987

1. The problem statement, all variables and given/known data

let's define {b_n} as an infinite sequence such that
all terms are nonnegative, and
for any i, j >= 1,

b_{i+j} <= b_i + b_j

it's easy to show that b_n/n <= (b_1) for all n, but
how would you show that
{(b_n)/n} is a convergent sequence?

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

Using induction, I got upto 0 < = b_n / n < = b_1
but that was pretty much it.
I have to show that it's monotone from some N on or something like that, i think..
but I'm just stuck about how to show that.

Last edited: Sep 19, 2010
2. Sep 19, 2010

Dick

If you've shown b_n<b_1/n then b_n/n<b_1/n^2. Can you show b_1/n^2 is a convergent sequence? An integral test should work.

3. Sep 19, 2010

evalover1987

my bad, problem fixed. i meant b_n < = b_1