Hey guys,

I'm having a little trouble with these two questions. It would be great if you could lend me a hand

p1.

$\displaystyle \cup_{k=1}^{\infty} [1 +\frac{1}{k}, 5 - \frac{1}{k}] = (1 , 5) $

By the archimedian property, there is some n $\displaystyle \in $ N such that, $\displaystyle \frac{1}{n} < \epsilon $

Therefore, $\displaystyle \cup_{k=1}^{\infty} [1 +\frac{1}{k}, 5 - \frac{1}{k}] \geq \cup_{k=1}^{\infty} [1 +\epsilon, 5 - \epsilon] \geq (1,5) $

(1,5) $\displaystyle \geq \cup_{k=1}^{\infty} [1 +\frac{1}{k}, 5 - \frac{1}{k}] = (1 , 5) $

I don't understand why $\displaystyle \cup_{k=1}^{\infty} [1 +\epsilon, 5 - \epsilon] \geq (1,5) $

Also why does (1,5) $\displaystyle \geq \cup_{k=1}^{\infty} [1 +\frac{1}{k}, 5 - \frac{1}{k}] = (1 , 5) $ ?