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**Danneedshelp** Consider a convergent sequence $\displaystyle (x_{n})_{n}$ such that $\displaystyle limx_{n} = x$.

(a) Let $\displaystyle (y_{n})_{n}$ be built such that $\displaystyle y_{n} = x_{n}+100$ , for all $\displaystyle n\in\mathbb{N}$. Is $\displaystyle (y_{n})_{n}$convergent? If yes, what is the limit?

(b) Let $\displaystyle (z_{n})_{n}$ be now the sequence $\displaystyle \{0, \pi\\, e, -\sqrt{2}, ln(2), x_{1}, x_{2} , x_{3} , . . . , x-{n} , . . . \}$. In other words, we added 5 extra terms at the beginning of $\displaystyle (x_{n})_{n}$ and we called this new sequence $\displaystyle (z_{n})_{n}$ . Is $\displaystyle (z_{n})_{n}$convergent?

If yes, what is the limit?

(c) Consider now a convergent series $\displaystyle \sum_{

n=1}a_{n}$ . Fix some natural number $\displaystyle K > 1$. Is $\displaystyle \sum_{n=K}a_{n}$ convergent?

(a) Assume converges to x and let $\displaystyle y_{n} = x_{n}+100$. Since $\displaystyle (x_{n})_{n}$ is convergent, $\displaystyle (x_{n})_{n}$ has convergent subsequences. Moreover, these subsequence congerge to the same limit as $\displaystyle (x_{n})_{n}$, $\displaystyle x$. Cleary, $\displaystyle y_{n} = x_{n}+100$ is a subsequence of $\displaystyle (x_{n})_{n}$. Thus, $\displaystyle (y_{n})_{n}$ converges to $\displaystyle x$.

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 $\displaystyle x$? 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