Local finiteness is restricted direct product-closed

This article gives the statement, and possibly proof, of a group property (i.e., locally finite group) satisfying a group metaproperty (i.e., restricted direct product-closed group property)
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Statement

Suppose $I$ is an indexing set and $G_i,i\in I$ is a collection of locally finite groups, then the restricted direct product of the $G_i$s is also locally finite.