Final answer:
A neighborhood T of a point x includes an open interval around x, which means there exists an epsilon (ε) such that (x - ε, x + ε) is contained within T. By this definition, we can prove such an ε exists and that (x - ε, ε + x) is a subset of T.
Step-by-step explanation:
To prove there is an ε (epsilon) > 0 such that the interval (x - ε, ε + x) is a subset of T, where T is a neighborhood of x, we proceed as follows:
By definition, a neighborhood of a point x in the context of real numbers is a set that contains an open interval around x. This means that there exists some ε > 0 such that (x - ε, x + ε) is entirely contained within the neighborhood T. Thus, it is guaranteed by the definition of a neighborhood that such an ε exists.
To show this formally, since T is a neighborhood of x, there must exist some δ > 0 such that (x - δ, x + δ) ⊂ T. We can simply take ε = δ, which proves that (x - ε, ε + x) is indeed a subset of T.