Let $n \geq 2$, and let $U_n^{\mathrm{up}}(\mathbb R)$ denote the group of all upper triangular matrices in $GL(n,\mathbb R)$ whose diagonal entries are all equal to $1$. For subgroups $H,K\leq U_n^{\mathrm{up}}(\mathbb R)$, let $[H,K]$ denote the subgroup generated by all commutators $hkh^{-1}k^{-1}$ with $h\in H$ and $k\in K$. Define the lower central series by $\gamma_1(U_n^{\mathrm{up}}(\mathbb R))=U_n^{\mathrm{up}}(\mathbb R)$ and $\gamma_{k+1}(U_n^{\mathrm{up}}(\mathbb R))=[\gamma_k(U_n^{\mathrm{up}}(\mathbb R)),U_n^{\mathrm{up}}(\mathbb R)]$ for every integer $k\geq 1$. Then $\gamma_n(U_n^{\mathrm{up}}(\mathbb R))=\{I\}$. In particular, $U_n^{\mathrm{up}}(\mathbb R)$ is nilpotent, with nilpotency class at most $n-1$.