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The magnesium strip burns brightly in the air, but continues to burn in the carbon dioxide environment.
This reaction is a combustion and oxidation. What's unusual, however, is that magnesium is reactive enough to be combusted and oxidized in a reaction with carbon dioxide:
2 Mg + CO2 -> 2 MgO + C
Under normal combustion/oxidation circumstances, oxygen is the reactant.
Carbon dioxide fire extinguishers work by smothering a fire in carbon dioxide. This is only an effective means of extinguishing a fire if carbon dioxide itself cannot be used as a fuel source.