Can a Sequence with a Limit of p be Called Infinite?

[amanda_ou812]

Homework Statement

If I have a sequence {Pn} and I know that lim Pn = p, can I call {Pn} infinite? I am trying to use this result in a real analysis proof. I know B(p; r) intersection S is non-empty and I need to show that it has indefinitely many points. I can show that {Pn} is a subset of S and is also a subset of B(p;r). So, if {Pn} is infinite, then B(p;r) intersection S would have indefinitely many points. Our definition of {Pn} is not strictly defined. Just that n is a natural number. I know that sequences can be finite or infinite but I am not sure of the definitions. Thanks!

 

[SammyS]

Do you have a definition for the limit of a sequence?

 

[amanda_ou812]

We are using the definition: Let (X, d) be a metric space, {pn} is a subset of C and is a sequence in X and p is an element in X. We say that the sequence converges to p and write lim (as n-> infinity) pn = p provided that for every e>0, there is a real number N so that when n>N the d(p, pn)<e

 

[amanda_ou812]

Sorry, {pn} is a subset of X

 

[SammyS]

We are using the definition: Let (X, d) be a metric space, {pn} is a subset of C and is a sequence in X and p is an element in X. We say that the sequence converges to p and write lim (as n-> infinity) pn = p provided that for every e>0, there is a real number N so that when n>N the d(p, pn)<e

You wrote, "lim (as n-> infinity) pn = p". Doesn't that imply that if a sequence does converge to a limit, then it must be an infinite sequence ?

Added in Edit:

Of course, just because a sequence is infinite, it doesn't follow that all of its terms are distinct.

 

[amanda_ou812]

hm...I see your point. So, {pn} infinite and {pn} as a subset of B(p; r) intersection S does not imply that B(p;r) intersection S has indefinitly many points. Is there another way I can get to that conclusion?