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I'm not sure where to start with this. I was thinking of assuming 1 is a limit and arriving at a contradiction?

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    that is the easiest approach! You just have to write down the definition of limit and plug in 1 as the limit. Show the definition doesn't hold! – 2017-01-22
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    @RSerrao Oh true, that's probably best. – 2017-01-22
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    You mean you want to show that $\lim_{n\to\infty}\frac1n\neq1$? Or what? – 2017-01-22
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    Or you could just show that $0$ is a limit of $1/n$ and quote the uniquness of the limit (in a metric space) – 2017-01-22

2 Answers 2

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$1$ is not a limit point of the sequence $\frac{1}{n}$ because $|1-\frac{1}{n}|\geq 1/2$ for all $n\geq 2$. So taking $\epsilon=\frac{1}{2}$ there is no $N$ such that $|1-\frac{1}{m}|<\epsilon$ for all $m\geq N$.

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Given $\epsilon=1/2$ we have that $$|1-n^{-1}|\geq\epsilon$$ for $n\geq 2$.