Hey guys, after spending a few hours on this question and not getting anywhere, I think I need some help.

Question

$\displaystyle \bigcap_{n \in N} A_n $ denotes the intersection of the sets $\displaystyle A_n$ and $\displaystyle n \in N $ and $\displaystyle \bigcup_{n \in N} A_n $ denotes the union of the sets $\displaystyle A_n $ and $\displaystyle n \in N $. ($\displaystyle N$ is the set of Natural Numbers)

Prove that $\displaystyle (0, \frac{4}{3} ] = \bigcup_{n \in N} \left[ \frac{1}{n+2} , 1 + \frac{1}{n+2} \right] $

and prove that $\displaystyle \{1\} = \bigcap_{n \in N} \left[1, 1+ \frac{1}{n} \right]$

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We have been learning about the Archimedean property, so I've been trying to mess around with that. I mean the second one is fairly obvious. 1 will be included in the set for any n. And therefore it is the only number than is always in the set of natural numbers and the set [1, 1+ 1/n]...but I don't know how to write this formally.

Regarding the second one, again, I can see that 1/(n+2) tends to 0 as n tends to infinity. Perhaps I need to prove this via the archimedean property? But I don't think that'd be enough. 4/3 is the highest that 1+ 1/n+2 (n=1) could possibly be if n belongs to the set of natural numbers...but I'm just stating the obvious here.

Please help!