Suppose A is a non-empty subset of \mathbb{R} and \sigma is a lower bound forA. Prove that \sigma = \inf(A) if and only if for all \epsilon > 0, there exists x \in A such that x < \sigma + \epsilon.

Which of the following options correctly completes the statement?

A) If A is a non-empty subset of \mathbb{R} and \sigma is a lower bound for A, then \sigma = \inf(A) if and only if for all \epsilon > 0, there exists x \in A such that x < \sigma + \epsilon.
B) If A is a non-empty subset of \mathbb{R} and \sigma is a lower bound for A, then \sigma = \inf(A) only if for all \epsilon > 0, there exists x \in A such that x < \sigma + \epsilon.
C) If A is a non-empty subset of \mathbb{R} and \sigma is a lower bound for A, then \sigma = \inf(A) if and only if for all \epsilon > 0, there exists x \in A such that x > \sigma + \epsilon.
D) If A is a non-empty subset of \mathbb{R} and \sigma is a lower bound for A, then \sigma = \inf(A) if and only if for all \epsilon > 0, there exists x \in A such that x > \sigma + \epsilon.