Recent content by Perelman

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 :)

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##

let ##a = (a_1,a_2,a_3,a_k..)## be a zero-convergent sequence. then the sequence of sequences ##L_n = (a_1,a_2,...,a_n,0,0,0,...)## for n=1 to infinity in ##\ell## would converge to ##a## because ##d(L_n , a) = sup|L_n - a|##, all the elements up to and including ##a_n## turns to zero, therfor...

it will converges to ##a##, i think i found some kind of proof now. let ##a = (a_1,a_2,a_3,a_k..)## be a zeroconvergent sequence. then for all ##n,m > N , d(a_n, a_m)<\epsilon##, for some ##N## then let ##n## be ##n+1## and ##m## be ##n+a## s.t ##d(a_{n+1}, a_{n+a})<\epsilon.## then the...

i still don't see it. what properties does this sequence (a1,a2,a3...) have ? is it like a special kind of general structure of all these zero convergent sequences. Because i don't see anything general about those two concrete examples you gave me or how they are relevant. I'm sure there are...

well if ##u_k = (1/2^0, 1/2^1 , 1/2^2 , 1/2^3, 1/2^4 ... 1/2^k...0,0,0, ... )## then the sequence ##u_k## from k=0 to infinity will converge to that ?

i can't find anything about it so maybe zero sequence could be part of it.

i don't get it since if I am not mistaken then there is lots of more sequences that converges to zero, and the sequence in L that converges to it would probably not look like post #1 sequence at all. i might have some holes in my understanding of this so don't hesitate by giving me details that...

yea the sequence from post #1 would work for that concrete example

it seems like you know some standard way of doing these things, i have no idea how to really construct such a sequence.

ye it gets a bit complicated with my notation when its sequence of sequences. well i proved that there can't be a sequence in ##\ell## that converge to a non-zero-limit sequence. so then all sequences in ##\ell## must converge to a zero-limit sequence. if I am not mistaken. so then that's...

but i think that's what i was trying to say in the last post so that for all ##n>N## for some natural number N then - the sequences ##x## will only now have infinite trailing zeros forever - while ##u## will have something bigger than ##\epsilon## forever. so when u are doing distance between...

hmm what about this? no sequenec x in ##\ell## can satisfy ##d(u,x) < \epsilon## because there exist an N s.t for all ##n>N , d(x_n,0) < \epsilon##, but ##d(u_n,0) > \epsilon##, and then for all ##n>N, |u_n - x_n| > \epsilon## and ##sup|u - x|## must also be ##> \epsilon##, making ##d(x ,u) >...

ok so since its a Cauchy sequence at some point you have ##d(x_n, x_m) < \epsilon## where ##x_m## is the sequence that's non-zero convergent, call it ##u_k## then the ##sup|x_n, u_k|## where ##x_n = {x_0, x_1, x_2...x_i, 0,0,0...}## with infinite zeros and ##u_k = {u_0, u_1...u_k}## for k = 1...

well negating the statement then i get if a sequence doesn't converge to 0 then there exist some ##n>N## such that ##d(x_n, 0) > \epsilon##. and then some contradiction happens i guess. but i can't see this.. especially if it supposed to be easy. so i need to use the fact that its also a Cauchy...