How does (*) follow from any of the above? What you're saying is true (that if a convergent sequence is always non-negative, then its limit is non-negative), but it's equivalent to what you're trying to show. And I don't see how it follows from what you've written. (What have you done with , apart from summoning them into existence?)

A proof by contradiction works well. Suppose that and that . Suppose that . Take small enough that . What do you get if you find large enough that for ? (Break up the absolute value into two inequalities; one of them will contradict the non-negativity of ).

Then take to be the sequence , as you did.