We are given the
Statement: If $A$ is a real square matrix with all real eigenvalues, then $A$ is orthogonally similar to an upper triangular matrix.
Now let $A$ be as given above.
Consider $A^T$; it too is a real square matrix, and since its eigenvalues are the same as those of $A$, we see they are all real, and thus there is an orthogonal matrix $O$ and an upper triangular matrix $\Delta$ such that
$A^T = O\Delta O^T; \tag 1$
$A = (A^T)^T = (O \Delta O^T)^T = (O^T)^T \Delta^T O^T = O \Delta^T O^T; \tag 2$
we thus see the matrix $A$ is similar to the lower triangular matrix $\Delta^T$ by the same orthogonal transformation $O$.