Consider a convergent sequence

such that

.

(a) Let

be built such that

, for all

. Is

convergent? If yes, what is the limit?

(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?

Thank you