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Abstract: When a finite statement in discrete maths is being extended to its infinite analogue, a useful method for brushing details under the rug is to say that we "use a standard compactness argument". In this talk, I will try to outline the basic structure of these arguments using many examples, give some idea for the underlying machinery (the Compactness Theorem from first-order logic), and if time permits – give some settings in which these proofs don't work.