Is there a computably enumerable set $A$ such that every infinite subset of $A$ is noncomputable?
I think that it's a set $K = \{n\ \ |\ \ U(n,n) \text{ is defined}\}$ (which is noncomputable, $U(n,x)$ is universal computable function). But I can't understand, how to prove, that an arbitrary infinite subset of $K$ is noncomputable. What to do?