Prove that a set is Borel(and hence Lebesgue)

Question

I'm trying to practice for the real-analysis final exam and I found this...Could you please help?
For $n$ $\in$ $\mathbb{N}$, define the following subsets of $\mathbb{R}$: $$ A_n=\begin{cases} (0,1]\cup[n,n+1) & , n-even \\ (0,1]\cup[n,n+2) & ,n-odd \end{cases} $$ Justify why $A_n$ is Borel and find $\lim_{x \to +\infty} \lambda(A_n).$
I was thinking that we could write these intervals as unions of open intervals and, being countable, they are Borel, but I'm not sure if this is correct...Also, I think that the result of the limit is 2 in the first case and 3 in the last one?

Answer 1

All intervals are Borel sets and unions of two Borel sets are Borel. Hence each $A_n$ is Borel . As far as $\lim \lambda (A_n)$ is concerned the limit does not exist since there are two limit points $2%$ and $3$.