For any language $A$, there is $B$ such that $A \le _T B$ but $B \nleq _T A$ I am trying to come up with a proof for the following:


  For any language $A$, there exists a language $B$ such that $A \le_{\mathrm{T}} B$ but B $\nleq_{\mathrm{T}} A$.


I was thinking of letting $B$ be $A_{\mathrm{TM}}$, but I realize that not all languages are Turing reducible to $A_{\mathrm{TM}}$, so $A \le _T B$ would not hold.
What other choice of $B$ do I have that would allow me to write a TM which uses an oracle for $B$ to decide $A$?

Thanks!
Let $B=A'$, the Turing jump of $A$. This is a basic result in the theory of the Turing degrees.