Real analysis, sequence of sequences convergence proof

[Dick]

okay ill add that in in my paper. but is this the whole proof or do i also need the other things? since now $c_0$ could just be a boundary around $\ell$ but i guess that's also counted as $\ell$ being dense. maybe i also need to add proof that $\ell \subset c_0$

I think you've done all you need to do. But if you aren't convinced of that you should do more to convince yourself.

 

[Perelman]

I think you've done all you need to do. But if you aren't convinced of that you should do more to convince yourself.

but i mean we did three things

everything from $\ell$ is in $c_0$
all seqeunce in $\ell$ converger in $c_0$
and all "points" in $c_0$ has a seqeunce in $\ell$ that converge to it

so its all those together right?. also thanks a lot for your help :)

 

[Dick]

Sort of, we didn't prove all sequences in $\ell$ converge. We proved that if a sequence in $\ell$ converges, then it converges to a limit in $c_0$. You're welcome!