Consider a convergent sequence such that .
(a) Let be built such that , for all . Is convergent? If yes, what is the
(b) Let be now the sequence . In other words, we added 5 extra terms at the beginning of and we called this new sequence . Is convergent?
If yes, what is the limit?
(c) Consider now a convergent series . Fix some natural number . Is convergent?
(a) Assume converges to x and let . Since is convergent, has convergent subsequences. Moreover, these subsequence congerge to the same limit as , . Cleary, is a subsequence of . Thus, converges to .
Is it enough just to use a theorem to prove this one?
(b) Can I just note that the first 5 terms are constants and pull them out of the limit and conclude the sequence still converges to ? I am not sure how to formalize this. Do I need to use the definition of convergence or the cauchy definition to prove this?
(c) Not totaly sure how to approach this one. I would think that is iss convergent, because, even though K is some number latter on in the sequence, there will still be a rank for which the terms in the sequence are stuck in some epsilon neighborhood.
Can anyone help me with some formal proofs?