Convergent Sequences on l infinity

[Kreizhn]

Homework Statement

Define $R^\infty_f = \{ (t^{(1}),t^{(2}), \ldots, ) |\; t^{(i}) \in \mathbb{R}\; \forall i, \; \exists k_0 \text{ such that } t^{(k})=0 \; \forall k\geq k_0 \}$

Define $l^\infty = \{ (t^{(1}),t^{(2}), \ldots, ) |\; t^{(i}) \in \mathbb{R}\; \forall i, \; \sup_{k\geq 1} | t^{(k})| < \infty \}$

Observe that $R^\infty_f$ is a linear subspace of $l^\infty$. Show that $R^\infty_f$ is not closed in $l^\infty$, then show that the closure of $R^\infty_f$ is the space $c_0$;

Homework Equations

The space c_0 is the set of all sequences converging to zero

The Attempt at a Solution

It's not too hard to show that this set is not closed. It suffices to show that there is a convergent sequence in $l^\infty$ such that every term is in $R^\infty_f$, but whose limit is not in $R^\infty_f$. I constructed the following sequence

$x_1 = (1, 0, \ldots, )$
$x_2 = (1, \frac{1}{2}, 0 , \ldots, )$
$\vdots$
$x_n = (1, \ldots, \frac{1}{n}, 0, \ldots} )$

which converges to the point $a = (1, \frac{1}{2}, \ldots, \frac{1}{n-1}, \frac{1}{n}, \frac{1}{n+1}, \ldots )$

It's the closure part that I'm worried about. I'm not terribly sure how I would go about showing that...

 

[morphism]

It will suffice to show that every sequence in c_0 is a limit of sequences in R_f. Do you agree?

 

[Kreizhn]

Yes, since every point in c_0 will necessarily be the limit of some sequence in R_f. Though I think that this only shows that c_0 is a subset of the closure - not necessarily the whole closure.

 

[morphism]

Yes, but on the other hand, R_f clearly sits in c_0 (and c_0 is closed!).

 

[Kreizhn]

True enough.

So I need to show that every sequence in c_0 is a limit of sequences in R_f.

How do I show that every sequence that converges to zero is the limit of a sequence. Indeed, what does it mean for a sequence to be a limit of another sequence?

 

[morphism]

Think of this as a problem set in an abstract normed space. What does it mean for a sequence {x_n} to converge to x?