This is question from real analysis It may use the definitions of open(closed) sets and compact sets which are

Question

This is question from real analysis. It may use the definitions of open(closed) sets and compact sets which are

given below the question. Please show all the intermediate steps, thanks!

Open and closed subsets of R Recall that N(X,6) : {y | d(X,y) ; 6}. = (FL-£1 7 6)
Definition 5.11 VwE, E a 612*, Nlme) 2 E A subset E C R is called open if for all X E E there exists an e gt; 0 such that
N(X,E) Q E. A subset if called closed is its complement is open. The union of any collection of open sets is open. Proposition 5.1 Let A : {A,- | i E I} be a collection ofsets. If all the A,- are open, then UielA; is
open.
Math