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Everything you've said above seems to be correct @Itsokay - The oxygen goes from a 0 oxidation state in the reactants (on the left) to -2 in both of the products (on the right). So, its oxidation number has decreased, meaning it must have gained electrons.
Oxidation is Loss (of electrons), Reduction is Gain (of electrons) (OIL RIG)
So, O has been reduced. If something has been reduced, then another species in the reaction must have been oxidised (loss of electrons). Can you spot what that is? (Hint: Oxygen will nearly always have an oxidation state of -2 when bonded to a less electronegative element, except in the case of H2O2).