Let $U_1,U_2,\dots, U_k\in\mathbb R^{n\times n}$ be orthogonal matrices. Show that the product $U = U_1U_2\cdots U_k$ is an orthogonal matrix.
Note, $U=\prod\limits_{i=1}^kU_i = U_1U_2\cdots U_k$. Then we need to show that $U^TU=I=UU^T$. Showing $U^TU = I$, observe: \begin{equation} \begin{split} U^TU &= \left(\prod\limits_{i=1}^kU_i\right)^T\left(\prod\limits_{i=1}^kU_i\right) \\ &= \left(\prod\limits_{i=-k}^{-1}U_{-i}^T\right)\left(\prod\limits_{i=1}^kU_i\right) \\ &= \left(\prod\limits_{i=-k}^{-2}U_{-i}^T\right)\left(\prod\limits_{i=2}^kU_i\right) \\ &\vdots \\ &= U_k^TU_k \\ &= I \end{split} \end{equation} Now we show $UU^T=I$. Observe: \begin{equation} \begin{split} UU^T &= \left(\prod\limits_{i=1}^kU_i\right)\left(\prod\limits_{i=1}^kU_i\right)^T \\ &= \left(\prod\limits_{i=1}^kU_i\right)\left(\prod\limits_{i=-k}^{-1}U_{-i}^T\right) \\ &= \left(\prod\limits_{i=1}^{k-1}U_i\right)\left(\prod\limits_{i=-k+1}^{-1}U_{-i}^T\right) \\ &\vdots \\ &= U_1U_1^T \\ &= I \end{split} \end{equation} And hence, $U$ is orthogonal.
Is this a sufficient proof?