David Hume's problem of induction

Hume's problem for induction
1) If we are to justify the belief in any conclusion got by reasoning, then we will have to give some argument from premises to that conclusion.
2) Either the argument is deductive of inductive.
3) If the argument is deductive, then it is said to be unproblematic.
4) If the argument is inductive then it can be given either (a) a deductive reconstrual or (b) it remains inductive.
5) If (a) then this is unproblematic.
6) If (b) then we assume that we have a justification for the inductive proof.
7) But this is just what we are looking for, a justification of inductive inference; (6) assumes what we are to establish.
8) So the search for a justification either head off on an infinite regress, or it is circular in that it appeals to some other inductive form of reasoning.

Hume's conclusion
We must believe the conclusion of inductive inferences; but we lack a rational justification for believing the conclusion.