## 3 questions concerning open and closed sets for sets having to do with sequence space

I have 3 questions concerning trying to prove open and closed sets for specific sequence spaces, they are all kind of similar and somewhat related. I thought i would put them all in one thread instead of having 3 threads.

1) Given y=(y$_{n}$) $\in$ H$^{∞}$, N $\in$N and ε>0, show that the set A={x=(x$_{n}$)$\in$ H$^{∞}$:lx$_{k}$- y$_{k}$l<ε, for k=1,2,...N} is open in H$^{∞}$

2) Show that c$_{0}$ is a closed subset of l$_{∞}$ [Hint: if (x$^{(n)}$) is a sequence (of sequences!) in c$_{0}$ converging to x $\in$ l$_{∞}$, note that lx$_{k}$l $\leq$ lx$_{k}$ - x$^{n}_{k}$l + lx$^{n}_{k}$l and now choose n so that lx$_{k}$ - x$^{n}_{k}$l is small independent of k.]

3) show that the set B={x $\in$ l$_{2}$: lx$_{n}$l$\leq$1/n, n=1,2,..} is not an open set. [hint: is the ball B(0,r) a subset of B.]

for question 1
The metric for H^infinity is A,

d(x,y)=Ʃ$^{∞}_{i=1}$2$^{-n}$lx$_{n}$-y$_{n}$l

so if m is in A, then I can enclosed a ball B(m,$\delta$) of radius delta around m where $\delta$=M$_{k}$+1/2^k where M$_{k}$=max{lx$_{k}$-y$_{k}$l, k=1,..,N} would that work?

For question 2,

in the hint, where it says to choose n, so that.... how do i pick n so that lx$_{k}$ - x$^{n}_{k}$l is small.

I know that lx$_{k}$l≤ lxl, but you have a sequence x$^{n}_{k}$ converging to x, is d(x$^{n}_{k}$, x) ≥ d(x$^{n}_{k}$, x$_{k}$)?? If so, how can i use this fact to pick n?

for question 3.

From the hint where it asks whether the open ball of radius r around 0 is a subset of B. In this case, would i have to specific how large r would be so that the radius of B(0,r) would contains elements that are not in B. If so, how do i specific r. The minkowski inequality is not of any help.
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