Limits of Sequences Homework: Proving Limit of a_n/n = 0

[NickMusicMan]

Homework Statement

For a sequence a_n:

If lim (a_n) =2, use the definition of a limit to show that lim (a_n / n) = 0

all limits are as n goes to infinity

The Attempt at a Solution

I know that I need to show:

Give any $$\epsilon$$>0 there is some M so that

if n>M then |a_n / n| < $$\epsilon$$But I can't Figure out how to do that.

Any help would be much appreciated! I have an exam in a few days and I came across this as a practice problem.

-NN

 

[NickMusicMan]

by the way: I totally understand intuitively why this is the case, I just figure out how to express it formally.

I know that as n approaches infinity, the fraction approaches (a fixed number)/(infinity) , which means it approaches 0. How can I write this using the formal definition?

 

[vela]

You know the limit of a_n, so you know there exists an N such that for n>N, |a_n-2|<ε.

Can you tweak that to get something that'll bound |a_n/n| from above? Or maybe use the triangle inequality?

 

[NickMusicMan]

I have tried using the triangle inequality to do so, but i haven't figured anything out yet.

 

[Dick]

I have tried using the triangle inequality to do so, but i haven't figured anything out yet.

Pick a value of epsilon. Then just divide the formal definition of limit by n.