getting limit of the set $D_k=\{x:2-\frac{1}{k}Hi I am trying to find 
$\lim_{k\to\infty}D_k=\{x:2-\frac{1}{k}<x<2+\frac{1}{k}\}$. 
I am struggling though because at first I thought this is empty set $\emptyset$. 
But then I thought if I think of limit of $D_k$ as sequence of set, then I thought the answer might be $\{2\}$ since on the left side, the sequence of set always have 2 as its element and right side sequence will also have 2 as its element.
Could anyone kindly help me solve this problem, thank you.

Self Answer: Claim $\lim_{k\to\infty}D_k=\{x:2-\frac{1}{k}<x<2+\frac{1}{k}\}=\{2\}$.
For any k, 2 is contained in $D_k$. Thus it will be contained in $\lim_{k\to\infty}D_k=\{x:2-\frac{1}{k}<x<2+\frac{1}{k}\}$.
Let x be any point smaller than 2. By Archimedean property, we can choose k large enough s.t. $x<2-\frac{1}{k}$. Than x does not belong to $D_k$.
You can do this similar to any point bigger than 2.
Thus the claim is justified.
