Write down proofs of the following statements about sets \(A\) and \(B\) of real numbers:

(a) If \(x\) is an upper bound for \(A\), and \(x \in A\), then \(x\) is a least upper bound for \(A\).

(b) If \(A \subseteq B\), then a lower bound for \(B\) is also a lower bound for \(A\).

(c) If \(A \subseteq B\) and a greatest lower bound of \(A\) is \(x\), and a greatest lower bound of \(B\) is \(y\), then \(y \leq x\).