Thread: Another limit proof

Prove if lim(x sub n)=x and x>0 then there exists a natural number M such that x sub n > 0 for all n> or equal to M

Any hints on how to start this or what the end result should be would be greatly appreciated, thanks.

Originally Posted by hayter221

Prove if lim(x sub n)=x and x>0 then there exists a natural number M such that x sub n > 0 for all n> or equal to M

Any hints on how to start this or what the end result should be would be greatly appreciated, thanks.

if then for all there exists an such that:

for all

Now choose and you should find the result follows with

CB

I'm not sure I follow, what do I do with the epsilon...sorry I have just begun to learn this and I haven't fully grasped the idea of the limit definition.

Originally Posted by hayter221

I'm not sure I follow, what do I do with the epsilon...sorry I have just begun to learn this and I haven't fully grasped the idea of the limit definition.

You can choose anything you like for , that is the definition of convergence for a sequence.

What we do is choose a particular value to ensure that is positive from onwards. We do this by choosing any value of then as for all we have:



so:



rearranging:



for all

But , so:

for all

CB

thanks, much appreciated!