Let $(X, \mathfrak T)$ be topological space and suppose that $A$ is a subset of $X$. Then $Bd(A) \subseteq A'$.
My definition of boundary: Let $(X,\mathfrak T)$ be a topological space and let $A \subseteq X$. A point $x \in X$ is in the boundary of A if every open set containing $x$ intersects both $A$ and $X−A$.
My definition of limit point: Let $(X, \mathfrak T)$ be a topological space with $ A \subseteq X$. A point $x \in X$ is said to be a limit point of $A$ provided that every open set containing $x$ contains a point $A$ different from $x$.
I am in an introduction to proofs writing class. I have to decide if this is true or false and then prove or provide a counter example. I think this is true and so I have started a proof. Our proofs on topology always deal with sets and elements in sets so I am trying to mirror that here.
I would like to start with the following: Let $x \in Bd(A)$ then by the definition $x \in A$ and $x \in X-A$. Now would it be correct to continue with a proof or a do I need a proof by contradiction?