Let a b be a closed interval on the real line and c d an ope

Let [a, b] be a closed interval on the real line, and (c, d) an open interval which is a subset of [a, b]. Prove that the closed interval [c, d] is also a subset of [a, b].

Solution

(c,d) is an open interval. [c,d] is the smallest closed set containing (c,d) ie it is the closure of (c,d)

So, c,d are limit points of (c,d)

But, [a,b] is a closed set so it has no limit points outside [a,b]

HEnce, c,d are in the interval [a,b]

HEnce, [c,d] is subset of [a,b]