The space is compact and Hausdorff, therefore, normal. Cover it with a finite number of open sets $\{U_n\}$ such that each can be imbedded into $\mathbb{R}^{k_n}$ , let $\{f_n\}$ be the corresponding collection of imbeddings. By Theorem 36.1, there is a partition of unity $\{g_n\}$ dominated by $\{U_n\}$ . Similar to the proof of Theorem 36.2, we construct $F=(g_1, . . ., g_n, f_1\cdot g_1, . . ., f_n\cdot g_n):X\to\mathbb{R}^{n+k_1+ . . .+k_n}$ so that it is continuous and injective.