Prove by induction. Every partial order on a nonempty finite set has at least one minimal element.
How can I solve that question ?
One proof method (by induction) is on the size of your set.
The basis, a singleton. There is only one element so clearly it is minimal.
Assume the claim is true for sets of size $n$, now prove for $n+1$:
Either way, you found a minimal element in $A$ and your partial order.