Is it possible to provide a proof of some unsolved result using elementary methods? Is there no merit to this? Is it possible to provide a proof of some unsolved result using elementary methods? I get the feeling it would be looked down upon and/or not taken seriously. Why is this? 
Is there no merit to proving some conjecture using elementary methods?
 A: This text is reproduced from the preface of Ramsey Theorey (2nd edition) by Ronald L. Graham et al

The romanticized view of mathematics is that it proceeds in sudden bursts of brilliant insight.  Sometimes it happens just that way.  Van der Waerden's theorem, the central result of Ramsey theory, was proven in 1926.  As van der Waerden recalled:

After lunch we went into Artin's office in the Mathematics Department of the University of Hamburg, and tried to find a proof.  We drew some diagrams on the blackboard.  We had what the Germans call "Einf$\ddot{a}$l": sudden ideas that flash into one's mind.  Several times such new ideas gave the discussion a new turn, and one of the ideas finally led to the solution.
[van der Waerden 1971]

Van der Waerden's proof used a subtle double induction and when expressed quantitatively led to an extremely fast growing function.  Mathematicians --we three included-- searched for a different proof technique without these features.  In 1987 Saharon Shelah was shown van der Waerden's theorem and within a day or two found a new proof.  Whether Einf$\ddot{a}$lle or not, Shelah's proof avoids the double induction, involves only "reasonably" fast growing functions, and --best of all-- is totally elementary.

My combinatorics professor this semester was at a mathematics conference at the time when someone suddenly burst through the doors shouting "You all must see this!" holding aloft a copy of the newly published proof.  The conference took a halt while they excitedly examined and praised the newly found proof.

There can be places for respect and admiration for elementary proofs of both new and old results, assuming that it sufficiently adds to our current understanding.  If it is proving a result which already has an elementary proof, or if the proof is incorrect in some way, the community might not greet it so warmly, however I see no reason why that should be thought of as the norm.
