Suppose x is a real number and epsilon > 0. Prove that (x - epsilon, x epsilon) is a neighborhood of each of its members; in other words, if y (x - epsilon, x epsilon), then there is a delta > 0 such that (y - delta , y delta ) (x - epsilon, x epsilon).

See proof below

Step-by-step explanation:

We will use properties of inequalities during the proof.

Let . then we have that . Hence, it makes sense to define the positive number delta as (the inequality guarantees that these numbers are positive).

Intuitively, delta is the shortest distance from y to the endpoints of the interval. Now, we claim that , and if we prove this, we are done. To prove it, let , then . First, then hence

On the other hand, then hence . Combining the inequalities, we have that  , therefore as required.  

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