Complete by taking an arbitrary cauchy sequence

[gtfitzpatrick]

Homework Statement

(1) Prove the space \ell_\infty is complete
(2)In \ell_\infty(R) , let Y be the subspace of all sequences with only finitely many non-0 terms. Prove that Y is not complete.

The Attempt at a Solution

(1)I can show that \ell\infty is complete by taking an arbitrary cauchy sequence and showing that x_n\rightarrow x

(2)Im not sure how to go about this.I figure the best wat to show that its not complete is to try to prove that it is complete?but I am not sure where to start?

 

[Dick]

The best way to show it's not complete is to find a Cauchy sequence in Y whose limit is not in Y.

 

[gtfitzpatrick]

i have to find a sequence in R that doesn't converge in Y? I am not sure now to go about this?

 

[Dick]

i have to find a sequence in R that doesn't converge in Y? I am not sure now to go about this?

A sequence in Y whose limit is not in Y. I.e. a sequence of sequences with finitely many nonzero terms converging to a sequence that doesn't have that property. Think about it.

 

[gtfitzpatrick]

so i want a sequence with finitely many zeros but i want it to converge to 0 right?
0, 1, 1/2, 1/3, 1/4... converges to 0 which isn't in Y thus proving that Y isn't complete?
Is that really all i have to say?

 

[Dick]

One sequence is a POINT in Y. To talk about convergence in Y you need a sequence of points in Y. I.e. a sequence of sequences that converges to another sequence.

 

[gtfitzpatrick]

yes,sorry

(0,0,0,...)
(0,1,0,0..)
(0,1,1/2,0,0..)
(0,1,1/2,1/3,0,0..)
(0,1,1/2,1/3,1/4,0,...)

is this what you mean by a sequence of sequences?
Thanks again for all the help

 

[Dick]

That's it.