In a similar spirit to the dense+closed answer, there are some proofs where to show that a subset of a connected space is the whole space, one shows that it is non-empty, open and closed. An example of this is the proof that an a connected open subset of a connected metric space $\mathbb{R}^n$ is path-connected (the set of points you can meet from x is open and non-empty and its complement is open). Another is the proof of the identity theorem in complex analysis (the set where your two analytic functions have the same Taylor expansion locally and therefore agree locally is obviously open, and it is also closed because the set of points where the nth derivatives agree is obviously closed, and the set where they have the same Taylor expansion is therefore an intersection of closed sets).

In a similar spirit to the dense+closed answer, there are some proofs where to show that a subset of a connected space is the whole space, one shows that it is non-empty, open and closed. An example of this is the proof that an open subset of a connected metric space is path-connected (the set of points you can meet from x is open and non-empty and its complement is open). Another is the proof of the identity theorem in complex analysis (the set where your two analytic functions have the same Taylor expansion locally and therefore agree locally is obviously open, and it is also closed because the set of points where the nth derivatives agree is obviously closed, and the set where they have the same Taylor expansion is therefore an intersection of closed sets).