Let $A$ be a finite matrix game and assume its minmax is $0$: $$ \bar v=\min_{t\in\Delta(J)}\max_{i\in I} e_iAt=0. $$ Construct a sequence of rows $x_n$ of $A$ so that its Cesaro averages form a Blackwell sequence for the cone $$ C=\{x\in\mathbb R^J:x\ge 0\}. $$ The deterministic Blackwell sequence theorem then gives a limit mixed strategy $s\in\Delta(I)$ with $$ sAt\ge 0\quad\text{for all }t\in\Delta(J), $$ so player 1 can guarantee $0$.

References

• [MFoGT, Section 2.8, Exercise 4(2)] Laraki, Renault, and Sorin, Mathematical Foundations of Game Theory. Derives finite minimax theorem from deterministic approachability.