Let $ F $ be a normed vector space, and let $ U_{n - 1},\dots,U_1,U_0\subset F $ be open subsets of $ F $. Let $ I\subset \mathbb R $ be an open interval.
Let $ \mathbf U = U_{n - 1}\times\dots \times U_1\times U_0 $, and let $ X\colon \mathbf U\times I\to F $ be continuous and locally uniformly Lipschitz vector field with respect to the second variable. I.e., suppose that for every $ \mathbf x_0 = (x_{n-1,0},\dots,x_{1,0},x_{0,0})\in \mathbf U $ and $ t_0\in I $ there exists neighborhoods $ M $ of $ \mathbf x_0 $ and $ N $ of $ t_0 $ where $$ \lVert X(\mathbf x,t) - X(\mathbf y,t) \rVert\leqq L_0\lVert \mathbf x - \mathbf y\rVert $$ for some $ L_0 > 0 $.
Define a new vector field $ Y\colon \mathbf U\times I\to F^n $ by $$ Y(\mathbf x,t) = (x_1,\dots,x_{n-1},X(\mathbf x,t)) $$ for any $ \mathbf x = (x_{n-1},\dots,x_1,x_0) $. I want to show that $ Y $ is again locally uniformly Lipschitz with respect to $ t $.
Take $ \mathbf x_0\in \mathbf U $ and $ t_0\in I $. There exists then some $ M $, some $ N $ and some $ L_0 $ as before. If now I compute (using the max-norm): $$ \begin{align} \lVert Y(\mathbf x,t) - Y(\mathbf y,t) \rVert &= \lVert (x_1,\dots,x_{n-1},X(\mathbf x,t)) - (y_1,\dots,y_{n-1},X(\mathbf y,t))\rVert\\ &= \max\{\lVert x_i - y_i\rVert,\lVert X(\mathbf x,t) - X(\mathbf y,t) \rVert\} \end{align} $$ I don't get to anything meaningful.
How can I show that $ Y $ has this useful property?