A Logic Problem

Consider this argument:

   1. A necessary truth is true.
   2. Whatever is true is possibly true.
   3. Whatever is possibly true could be false.
   Therefore
   4. A necessary truth could be false.
   
A sound argument is one that satisfies two conditions: its premises are all true, and the reasoning it embodies is correct. Is the above argument sound?

If not, what has gone wrong in the argument? Answer below the fold.

There is nothing wrong with the formal logic of the argument.  So if there is a mistake it is because of the falsity of one or more if the premises.  Premise 1 is plainly true.  A proposition true in every possible world is true in the actual world.  Premise 2 is also obviously true.  If a proposition is true, then it is possible that it is true:  if a proposition is true in the actual world, then it is true in at least one possible world.  Premise 3, however, is false: Every necessary truth is possibly true, but no necessary truth is false in any possible world.  And so the argument, having a false premise, is not sound.