The US isn't over stretched to wage a military campaign against North Korea. We have 80,000 Army personnal between S. Korea and Japan, we have our Flag Ship Fleet in the 7th Fleet, our Largest forward-deployed fleet in the Navy. The military personnal presence is by and large moot as it would be retarded to do a ground invasion of North Korea, in a conflict with them we would hold at the DMZ and push buttons (i.e. lots of missles and bombs).
Your question posed is a little weird in its wording, if NK took "a few shots"? I'm sure no amount of aggression would be tolerated; the thing is though, if aggression is taken, I doubt it will be limited to a few shots as myself and others have alluded to Seoul being pounded to shit by NK artillery.
And yes the rest of the world would no doubt cave in this instance.
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To win a war you must serve no master but your ambition.
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