Hello,

I am asked to show that the following two definitions of the limit \(\displaystyle L\) of a sequence \(\displaystyle s_{n}\) are equivalent:

a) \(\displaystyle \forall \epsilon>0 \exists N\in \mathbb{Z} \forall n \geq N: abs(s_{n}-L) < \epsilon\)

b) \(\displaystyle \forall m \in \mathbb{Z}_{+} \exists N\in \mathbb{R} \forall n \geq N: abs(s_{n}-L) < \frac{1}{m}\)

I do see it intuitively, but how do I show it quantitatively?

Thanks.

I am asked to show that the following two definitions of the limit \(\displaystyle L\) of a sequence \(\displaystyle s_{n}\) are equivalent:

a) \(\displaystyle \forall \epsilon>0 \exists N\in \mathbb{Z} \forall n \geq N: abs(s_{n}-L) < \epsilon\)

b) \(\displaystyle \forall m \in \mathbb{Z}_{+} \exists N\in \mathbb{R} \forall n \geq N: abs(s_{n}-L) < \frac{1}{m}\)

I do see it intuitively, but how do I show it quantitatively?

Thanks.

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