Existence of a pure-strategy Nash equilibrium - paper referring to a standard result

Question

I am reading a paper where the authors write the following regarding an $n$-player game where $u_i$ is the utility function of player $i$, and $q=(q_i)_{1\leq i\leq n}$ are the actions of each player:

"The existence of a pure-strategy Nash equilibrium follows readily from the concavity of the utility function $u_i(q)$ in $q_i$, its continuity in $q$, and the fact that the strategy spaces are compact. Since the arguments are standard, we omit a formal proof"

1. Can anyone tell me which standard result is referred to here?

2. Is there a go-to set of notes covering basic results for the existence and uniqueness of equilibrium solutions, and standard techniques to find such solutions?

Answer 1

I am a beginner, so it could be wrong, but I think they might be referring to the following theorem:

Consider a strategic-form game whose strategy spaces $S_i$ are non-empty compact convex subsets of a Euclidean space. If the payoff functions $u_i$ are continuous in $s$ and quasi-concave in $s_i$, there exists a pure-strategy Nash equilibrium.

This is from Game Theory by Drew Fudenberg and Jean Tirole, page 34. Very good book to have at hand and also use for studying!