Determine if the subset of R

$$ M := \{\frac{n-1}{n} : n \in \mathbb{N}, n\geq 1 \} \cup ]1,2] $$

M closed, compact or bounded is (proof) ?

Question

Determine  if the subset of R

$$ M := \{\frac{n-1}{n} : n \in \mathbb{N}, n\geq 1 \} \cup ]1,2] $$

M closed, compact or bounded is (proof) ?

experimentx · Answer

the set is bounded n=1 and n->inf, [0, 1) U [1, 2]

experimentx · Answer

the set is neither closed nor open. since you have [0, 1)

experimentx · Answer

I'm not sure about compactness ...

anonymous · Answer

thank you experimentX

experimentx · Answer

i tried my best.

anonymous · Answer

its good i think..  
can you explain why we got $$[0,1)$$ ?

experimentx · Answer

put n=1, you will get 0

put n-> infinity, you will get 0) ... since you can't put n=infinity directly.

anonymous · Answer

A compact set is closed and bounded.
Since it is not closed, then it is not compact.

anonymous · Answer

It is not closed since
$$
\lim_{n\to \infty}\frac{n-1}{n}=1
$$and 1 is not in the set.

anonymous · Answer

thank you very much mr Saab

anonymous · Answer

Mr @eliassaab  
when normaly a set closed and bounded, can you give me pls an example, so that i can understand it ?

anonymous · Answer

Like [0,1] is closed and bounded so it is compact.

anonymous · Answer

ok thank you