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If you use the word "trivial" in its usual meaning, then no proof is needed -- usually, "trivial" means "immediately obvious to any reader". But since you ask, it seems that you use the word in some other meaning -- perhaps in the meaning of "elementary", but this is only a guess. If this is so, then what I would do in this situation depends on the length of the proof, both in terms of pages as well as in terms of percentage of the total length of the paper, as well as on whether the lemma is needed in the proof of some key result of your paper or just in some example, etc.. If the proof does not make the paper significantly longer, or if it is not longer than one or two pages, I would certainly include it.