(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Define [itex] 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 \} [/itex]

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

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

2. Relevant equations

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

3. 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 [itex] l^\infty[/itex] such that every term is in [itex] R^\infty_f [/itex], but whose limit is not in [itex] R^\infty_f [/itex]. I constructed the following sequence

[itex] x_1 = (1, 0, \ldots, ) [/itex]

[itex] x_2 = (1, \frac{1}{2}, 0 , \ldots, ) [/itex]

[itex] \vdots [/itex]

[itex] x_n = (1, \ldots, \frac{1}{n}, 0, \ldots} ) [/itex]

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

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

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# Homework Help: Convergent Sequences on l infinity

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