Ordering relations smallest/minimal elements definitions In How to Prove It: A Structured Approach, 2nd Edition, page 192, the author introduces the following definitions of smallest and minimal elements of partial orders:

Definition 4.4.4. Suppose R is a partial order on a set A, B $\subseteq$ A, and b $\in$ B. Then b is called an R-smallest element
  of B (or just a smallest element if R is clear from the context) if
  $\forall$x $\in$ B(bRx). It is called an R-minimal element (or just a
  minimal element) if $\lnot$$\exists$x $\in$ B(xRb $\land$ x $\neq$ b).

Then the author introduces the following example:

Let L = { (x, y) $\in$ ℝ x ℝ | x ≤ y }, as before. Let B = { x $\in$ ℝ
  | x ≥ 7}. Does B have any L-smallest or L-minimal elements? What about
  the set C = { x ∈ ℝ | x > 7 }?

Solution:

Clearly 7 ≤ x for every x $\in$ B, so ∀x $\in$ B(7Lx) and therefore 7
  is a smallest element of B. It is also a minimal element, since
  nothing in B is smaller than 7, so ¬∃x ∈ B(x ≤ 7 ∧ x $\neq$ 7). There are
  no other smallest or minimal elements. Note that 7 is not a smallest
  or minimal element of C, since 7 $\not\in$ C . According to
  Definition 4.4.4, a smallest or minimal element of a set must actually be an element of the set. In fact, C has no smallest or
  minimal elements.

The part about B makes perfect sense to me, but I'm confused about:

In fact, C has no smallest or minimal elements

As far as my understanding goes, C does have an L-smallest element, which happens to be 8 (the example says that 7 is not a smallest/minimal element of C, which is obvious since 7 is not a member of C):
$$\forall x \in C(8Lx)$$
Which is obviously true since 8 ≤ 8, 8 ≤ 9, 8 ≤ 10, and so on.
Also, 8 looks like an L-minimal element of C as well, since:
$$\lnot \exists x \in C(xL8 \land x \neq 8)$$
The only element of C which is smaller or equal to 8 is 8, but 8 = 8.
Why does the author says that C has no smallest or minimal elements?

There is no real number which can serve as a minimal element. 8 is the smallest natural number.
