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So I'm putting some final touches to my first draft and I notice that I've written a few clauses ending with prepositions.

I remember I had this one old-school professor who insisted that they be removed/relocated, but I've always been in the camp that it's an acceptable usage if it feels natural. I did some research on the Internet and it seems that the majority of grammarians today agree with my position regarding this issue, so I'm tempted to go with the sentences that I have but I'm also worried that there might be an adcom who's a stickler for this no-preposition "doctrine" (I refuse to call it a rule because there's no real justification for it in the English language and grammarians are leaning against its favor).

tl;dr: clause ending with a preposition in PS - yay or no-no?

A random side note: Supposedly, an editor once corrected Churchill for this usage, which led Churchill remark "This is the kind of tedious nonsense up with which I will not put." lulz.

I'd suggest erring on the side of being conservative. No one's gonna knock you for "correct" preposition usage, but someone might do so for "incorrect" usage, and anyone who stakes their "style" or whatev on such a minor thing is making a bad judgment in this context, imho. Churchill's thing is funny, but it's also hyperbolic and slightly inapt for a more formal context (ie., "put up with" is colloquial).