Show that for an infinite subset $\mathbb{M}$ in the space $\mathbb{S}$ to be compact, it is necessary that there are numbers $\gamma_1, \ \gamma_2,\dots$ such that $\forall x=(\xi_k(x))\in \mathbb{M}$ we have $|\xi_k(x)|\leqq \gamma_k$. (It can be shown that the condition is also sufficient for the compactness of $\mathbb{M}$)
I would be grateful if anyone could explain this exercise and told me how to solve it?