Answer to Question #264610 in Real Analysis for Nikhil Singh

Question

Show that the set [ -5,3] ∩[ -3,5] is a neighborhood of 2.

Answer 1

Let A = [-5, 3] ∩ [-3, 5]

We shall show that there exist an open interval which is a subset of A containing 2.

We know that A = [-5, 3] ∩ [-3, 5] = [-3, 3]

It is evident that (-3, 3) ⊂ [-3, 3]

Also, 2 ∈ (-3, 3)

Thus, (-3, 3) is an open interval which is a subset of A containing 2.

Hence, the set A = [-5, 3] ∩ [-3, 5] is a neighborhood of 2