Your proof is correct.
Can you check my relations proof? It feels correct but at the same it almost seems too easy so I'm wondering if perhaps I missed something.
Thanks in advance,
Suppose is a relation on a set with the property that for every for some , that is, every element is related to at least one other element in . Prove that if is symmetric and transitive, then is reflexive.
Let . By hypothesis, for some . Since is symmetric, . Since we have that and , and we know that is transitive, so that is reflexive.