Math Is Fun Forum

EMPhillips1989

Member

Registered: 2008-01-21

Posts: 40

convergent sequences

hey can anyone help me prove that the sequence {an} given by

converges to 0
this is clearly obvious i just dont see how to apply a proof can anyone please help???

Last edited by EMPhillips1989 (2008-02-04 08:10:15)

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: convergent sequences

By definition of convergence:

So basically, we want to prove that 1/N < ε, for any value that ε might take, as long as N is big enough (because if n>N then 1/n <1/N and so 1/n < ε).

Rearranging this gets that N > 1/ε.
So as long as an N exists such that it's bigger than 1/ε, then 1/n does indeed converge to 0.

The Archmides Principle says that given any real number, a natural number exists that is bigger than it. Kind of obvious, but it's the fact we need here.

So then you set N equal to this number and that proves convergence.

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Why did the vector cross the road?
It wanted to be normal.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: convergent sequences

Just to tidy up some loose strands in the statement of the definition:

There.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: convergent sequences

Sorry. I was concentrating so much on getting my LaTeX sorted that I didn't check if it said what I wanted it to.

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Why did the vector cross the road?
It wanted to be normal.