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## Homework Statement

Show that ##\displaystyle \bigcup_{n=2}^\infty \left[ \frac{1}{n} , \frac{n}{n+1} \right] = (0,1)##.

## Homework Equations

## The Attempt at a Solution

I'm not sure how to show this rigorously. It is sufficient to note that ##\lim_{n\to\infty} \frac{1}{n} = 0## and that ##\lim_{n\to\infty}\frac{n}{n+1} = 1##? How can I verify that this union actually isn't ##[0,1]##?