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Problem 41 (More Practice With Universal Quantifiers)

For each $n\in \mathbb{N}$, define $\ds a_n = \frac{n-1}{n}$. Only one of the statements below is true. Rewrite each statement using the quantifiers $\forall$ and $\exists$. Then determine which statement is true, prove that statement is true, and then prove the other statement is false.

  1. For each real number $\varepsilon>0$, there exists $N\in \mathbb{N}$ such that for every $n\in \mathbb{N}$, if $n>N$ then $\ds \left|a_n-0\right|<\varepsilon$.
  2. For each real number $\varepsilon>0$, there exists $N\in \mathbb{N}$ such that for every $n\in \mathbb{N}$, if $n>N$ then $\ds \left|a_n-1\right|<\varepsilon$.

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