A real number $x$ is called nearly-even, if there is an even integer $n$ such that

$| x - n | < 0.5.$

If $X$ is a nearly-even number, which of the following statements must be true?

$\begin{array}{r r l} & \text{I.}\ & X\ \text{is an even integer.}\\ & \text{II.}\ & 6X\ \text{is an integer.}\\ & \text{III.}\ & \text{The integer part of}\ 6X\ \text{is even.}\\ \end{array}$

(A) $\ \$ II only
(B) $\ \$ I and II only
(C) $\ \$ I and III only
(D) $\ \$ II and III only
(E) $\ \$ None of the statements