Should I cite a result if the paper doesn't include a proof?

Citing the other paper seems necessary in any case, as they have stated the lemma before you. This is for attribution. Citing the other paper for evidence seems not appropriate, as there is no proof given there.

If the lemma is not rather obvious (say the obvious proof strategy works in < 5min), then stating it without proof would be very bad form. Put in an appendix if you don't want a boring lengthy proof to spoil the otherwise elegant paper, but put it somewhere people can find it.

If the result is basically trivial (as you say it is), I think how you proceed should consider how standard this type of result would be in the field.

You could put something like:

The following result can be established by standard (but tedious) computation.

if it's the sort of thing you could expect an early graduate student in the field to do as a homework question, or

The following result, which is stated by (Author) in [(paper)], can be established by standard (but tedious) computation.

otherwise.

Cite the paper when you state the lemma. Then write:

\begin{proof} Split up the domain of integration to remove absolute values, then evaluate each of the parts. \end{proof}

It's a waste of everyone's time to have two pages of a calculus exercise. But it's also a waste of everyone's time to have to guess how the proof goes. The above is the best compromise that makes it clear how the proof goes in the least amount of time.

If the proof were one paragraph instead of two pages then I'd say include it all.