Existence of solution to maximization problem I'm a first year grad student and we're learning about the utility maximization problem and the producer's problem now. We have been assuming that a solution to these problems exists. But every now and then there's an exercise problem asking us to show that a solution exists. I have encountered once that "A maximization problem on a compact set has a solution". I wonder if there's any good reference on this? We're using MGW's Microeconomic Theory as our textbook; I thought I could find something in the appendix but I failed.
It is most likely that the statement is in the form of 'A utility maximization problem on a compact set has a solution when the utility function is continuous.' In the one dimensional case, this is a direct corollary of the extreme value theorem. A reference to this theorem can be found at https://en.wikipedia.org/wiki/Extreme_value_theorem 

However, discontinuity of the utility function may break this result. Consider the following function.
$$
u(x) =  \begin{cases} x \mbox{ if $x\in[0,\frac{1}{2})$,} \\ \frac{1}{4} \mbox{ if $x \in [\frac{1}{2},1].$   } \end{cases} 
$$
It should be immediate to recognize that this function does not have attain a maximum for any $x\in [0,1]$ even though we have a compact set.

A book that I had used before is here which deals with optimization problems of the sort of an economist would encounter in a rigorous way. Also, this book provides a more intuitive way of establishing the main results, which would be a better start.