How do I prove designed Min Interval Algorithm?

Question

I am trying to learn how to prove the correctness and optimality of algorithms. For this I am trying to do it for the problem:

Input is interval list ([l1,r1],[l2,r2],...,[ln,rn])

Find the minimal interval coverage that is the coverage with the least amount of intervals containing all intervals from the interval list.

For ([1,4],[2,7],[10,11]) it would be ([1,7],[10,11])

sort(X)  //sort X by increasing interval start l
l' <- left(x0)
r' <- right(x0)
remove(x0,X)  //remove x0 from interval list X
for x in X:
 l<- left(x)
 r<- right(x)
 if r>r' and l<=r'+1:
  r'<-r
 if l>r':
  expandedX = [l',r']
  Result.add(expandedX)
  l'=left(x)
  r'=right(x)

Runtime in O(nlogn+n)

Now I really don't know how to prove it.

How do I get to an systematic approach resulting in a valid good proof?

My attempt:

Let Y be the interval coverage produced by the algorithm and let Y' be the optimal interval coverage.

Suppose Y is not optimal.

Hence |Y'|< |Y|

This implies that there exists an y in Y such that y not in Y' and there exists y' such that y is a subset of y'

This implies that there exists y'' in Y such that left(y'')<=right(y)

Is this a valid start?