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Many books, particularly in mathematics and related disciplines, contain sentences like "the proof is left as an exercise to the reader", which actually mean that the claim should be easy to prove for anyone reading until that point.

Can I write such a sentence in a paper submitted to a journal? Or will the reviewers consider it rude to omit a proof in such a way?

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    It is not rude. If your fear is that the word "exercise" could be seen as condescending to the reader, then write simply "the proof is left to the reader". Personally I think it communicates more to give some, albeit brief, indication of how to prove the claim in question, even if it is just to say "the proof is is similar to the proof of claim X" or "the claim is proved by applying Lemma Z and the known fact that Blah Blah". A reviewer will only be bothered by such a phrase if the omitted proof is really not so self-evident as you imply.
    – Dan Fox
    Dec 31 '14 at 12:51
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    @DanFox: Please consider converting your comment into a full-fledged answer.
    – aeismail
    Dec 31 '14 at 13:18
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    For what it's worth, I have heard of people using such a statement because they themselves have not figured out the proof yet and need more time to do so. Dec 31 '14 at 13:44
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    a real example : stackoverflow.com/questions/12244968/…
    – Offirmo
    Dec 31 '14 at 15:02
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There are two separate issues here: omitting the proof, and calling it an exercise.

It is relatively common to omit some proofs in research papers. In particular, proofs that are long but routine, or that re-use the same method as other proofs in the paper, are often abbreviated or removed. It depends on the intended audience for the paper, to some extent. You will get a sense of this by reading more research papers. But caution is needed: one of the first places to look for errors in a paper is in results where the proof was omitted!

The question whether to call something in your paper an "exercise" is more difficult. I wouldn't be surprised if some people do it -- but some people don't write very well. The audience for a paper includes your peers, and you are not really in a position to assign them homework. There are other ways to indicate that a proof is easy besides calling it an exercise: "the following theorem follows directly from definitions", "the proof of the following theorem is routine", "the proof of the following theorem is similar to the proof of Theorem 1.3", etc. Giving a small hint of how to prove the result can make the paper stronger, and it helps any readers who are not experts in the area.

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I think it depends on the context: If it is a research paper, it is definitely rude. As a referee, I would return such a paper to the author to fill it in. If it is a review paper or aims at education in math then it’s fine in my opinion. A different story are textbooks with or without exercises, where it is apropriate.

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