How to deal with results that are given as exercises I'm currently writing my thesis in mathematics and in the introductory part I want to give a brief overview over a field I'm not very familiar with. There are some results in the book I'm reading which are stated as exercises for the reader, which I'd like to include as theorems in my thesis. I don't want to prove them, since this part is only meant to give the reader an overview of some basic concepts and to show him the importance of some constructions which I will generalize later (which is the actual bread and butter of the thesis). So how do I deal with this:


Don't give any citation at all and assume these concepts are well known to anyone working in the field? - I wouldn't give the definition of a vector space in my thesis, so this might be justifiable. However, I wouldn't consider those theorems as basic as the definition of vector spaces...
Do a heavy amount of googling to find sone sources which prove those theorems? - This might take a considerable amount of time, so I'd like to avoid it. It might also turn out to be impossible, since the phrasing might be different enough in the original sources so that I just won't find them
Cite the Problem from the book I'm working with? - I don't think I can do this, since the results aren't proved in the reference I would give


Note that none of my results rely on any of these theorems.
In mathematics, if you cite a result, you should really be able provide the proof if somebody asks for it. So I would strongly recommend against option 3.

One thing you might consider is writing the author of the textbook, explaining your situation, and asking if he's willing to send you a proof. Then you can cite it as 


  problem 8.35 in textbook, proof provided by private communication with [author]


even if he would rather not have a proof of the exercise published. 

Of course, if he replies "I think I had a proof when I wrote the textbook, but I've forgotten it now," you might need to come up with a proof in some other way.

There is at least one exercise in a textbook whose proof is highly non-trivial, and was an open problem for several years before a journal paper giving the proof was published (and the only indication in the textbook of this is that part (b) of the problem is "optional").