1. ## Question about converging sequences.

I know that a converging sequence a(n) to an a∈R is bounded. (I am pretty sure this is proven.) Is the above also valid for sequences converging to 0, i.e. Nullsequences -- Zerosequencies?

If yes, can this be proven somehow with an equality or any other way? If yes, how?

thank you

2. ## Re: Question about converging sequences.

Originally Posted by nappysnake
I know that a converging sequence a(n) to an a∈R is bounded. (I am pretty sure this is proven.) Is the above also valid for sequences converging to 0, i.e. Nullsequences -- Zerosequencies?
Of course is also valid. Hint Choose $\displaystyle \epsilon =1$ , if $\displaystyle \lim_{n\to \infty}a_n=0$ there exists $\displaystyle n_0\in \mathbb{N}$ such that $\displaystyle |a_n|<1$ if $\displaystyle n\geq n_0$ that is, all terms of the sequence are in $\displaystyle (-1,1)$ except for a finite number.

3. ## Re: Question about converging sequences.

amazing, thank you soo much Fernando for the explanation!