- #1
Paalfaal
- 13
- 0
Homework Statement
Define:
- c0 = {(xn)n [itex]\in[/itex] [itex]\ell[/itex][itex]\infty[/itex] : limn → [itex]\infty[/itex] xn = 0}
- l0 = {(xn)n [itex]\in[/itex] [itex]\ell[/itex][itex]\infty[/itex] : [itex]\exists[/itex] N [itex]\in[/itex] the natural numbers, (xn)n = 0, n [itex]\geq[/itex] N}Problem: Prove that [itex]\overline{\ell}[/itex]0= c0 in [itex]\ell[/itex][itex]\infty[/itex]
Homework Equations
The Attempt at a Solution
I want to find the solution using the limit-definition of closure.
Considering an element
x = (xn) [itex]\in[/itex] c0
and a sequence
yj = (xjn) [itex]\in[/itex] [itex]\ell[/itex]0,
such that xnj = xn for n < j, xnj = 0, otherwise.
Using the metric induced by the supremum norm; || xn - xnj ||∞ [itex]\rightarrow[/itex] 0 as j tends to infinity. We can du this for all elements in c0, and hence c0 [itex]\subseteq[/itex] [itex]\overline{\ell}[/itex]0.
My problem is to show the other direction, that is
[itex]\overline{\ell}[/itex]0 [itex]\subseteq[/itex] c0
I need to show that elements in [itex]\overline{\ell}[/itex]0 has a vanishing limit. I don't know how to do this using the supremum norm. In fact, it seems impossible to me..
Can I get any help?