$GL(n)$ is the group of all invertible $n \times n$ matrices and $O(n)$ are all orthogonal $n \times n$ matrices.
I have to prove that $O(n) \subset GL(n)$.
I can prove that by showing that the neutral element of $GL$ is in $O$ and that the operations set of $O$, matrix multiplication and matrixinversion, are under closure.
How do I start with the second part?
A matrix is orthogonal if and only if its inverse is its transpose, since they preserve inner (dot) products: $(x,y)=(Qx,Qy)$, which implies that $(x,Q^{-1}y)=(Qx,QQ^{-1}y)=(Qx,y)$. On the other hand, if $(Qx,y)=(x,Q^{-1}y)$, then we have $(Qx,Qy)=(x,Q^{-1}Qy)=(x,y)$.
Hence, $QQ^{T}=Q^{T}Q=I$, so inverses are okay.
On the other hand, for any $A,B \in O(n)$, that $(AB)^T=B^TA^T$, so we have that $AB(AB)^T=I$ by basic calculation