Finitary symmetric group on infinite subset is conjugate-dense

Statement
Suppose $$S \subseteq T$$ are infinite sets. Let $$\operatorname{FSym}(S)$$ and $$\operatorname{FSym}(T)$$ denote the finitary symmetric groups on the sets $$S$$ and $$T$$ respectively, with $$\operatorname{FSym}(S)$$ viewed as a subgroup of $$\operatorname{FSym}(T)$$: any finitary permutation of $$S$$ is extended to a finitary permutation of $$T$$ by simply fixing all points in $$T \setminus S$$. Then, $$\operatorname{FSym}(S)$$ is conjugate-dense in $$\operatorname{FSym}(T)$$: any finitary permutation on $$T$$ is conjugate, in $$\operatorname{FSym}(S)$$, to a finitary permutation on $$S$$.

Related facts

 * Symmetric group on proper subset is not conjugate-dense
 * Symmetric group on finite or cofinite subset is conjugacy-closed
 * Symmetric group on infinite coinfinite subset is not conjugacy-closed
 * Finitary symmetric group on subset is conjugacy-closed