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For a sequence $\left \{ a_{n} \right \}$ of real numbers, which of the following is a negation of the statement ‘$\underset{n\rightarrow \infty }{lim}\:a_{n}=0$’?

  1. There exists $\varepsilon > 0$ such that the set$\left \{ n\in\mathbb{N}\mid\left | a_{n} \right | > \varepsilon \right \}$ is infinite. 
  2. For any $M>0$, there exists $N \in \mathbb{N}$ such that $\left | a_{n} \right |> M$ for all  $n\geq N$.
  3. There exists a nonzero real number a such that for every $\varepsilon > 0$, there exists $N \in \mathbb{N}$ with $\left | a_{n}-a \right |< \varepsilon$ for all $n\geq N$.
  4. For any $a \in \mathbb{R}$, and every $\varepsilon > 0$, there exist infintely many $n$ such that $\left | a_{n}-a \right |> \varepsilon$.
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