Catching Kira: Going about it the wrong way? (Death Note)

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I was thinking about this since last week. Could it be that, in Death Note, that the NPA (Light), the SPK (Near), and the Mafia (Mello) are going about the wrong way to stop Light's misguided rampage? Look at it this way: One of the rules of the Death Note is that whenever a Shinigami's Death Note is dropped and falls to Earth, it's the property of whoever finds it at that time. Well since there's an indeterminate number of Shinigami and Death Notes in their world, it's not possible to limit the user to just one person. If possible, as we've seen, Death Notes can be transferred from one person to another. So there's the possibility that there can be more than one Kira. In that case, instead of spending all that time trying to catch the person using the notebook, wouldn't it be easier to find the source instead? Wouldn't it be simpler to find the Death Note and destroy it instead? That's what I was thinking though.