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I'm new to this website and thought it might help with my future mathematical studies. I'm trying show whether the problem $\lim_{n \rightarrow \infty} [0, 1 - 1/n] = [0, 1)$, $\lim_{n \rightarrow \infty} [0, 1 - 1/n) = [0, 1)$, $\lim_{n \rightarrow \infty} [0, 1 + 1/n] = [0, 1]$ and $\lim_{n \rightarrow \infty} [0, 1 + 1/n) = [0, 1]$. I tried using $[0, 1 - \lim_{n \rightarrow \infty} 1/n] = [0, 1)$ since $1/n = 0$ in the limit as $n$ goes to infinity. Is this the right approach? Also, I'm experiencing difficulties in understanding whether it should closed or open interval (i.e., $[0, 1)$ and $[0, 1]$) in the limit. I think this has to do with the limit inferior/superior but cannot put the logic together. These exercises were in sections of basic set theory and monotone sequence. Thank you.

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Hint:

Firstly, you should know the definition of $\lim_{n \rightarrow \infty} [0, 1 - 1/n] $ and so on.

In fact, for example, $\lim_{n \rightarrow \infty} [0, 1 - 1/n] =\bigcup_{1\le n} [0, 1 - 1/n] $.

Then only need to prove the two sets is equal.

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