The statement of the mean value inequality (MVI) is as follows: "Let A be an open convex subset of R^n and let f:A-->R^m be continuously differentiable and such that ||Df(x)(y)||<=M||y|| for all x in A and y in R^n (i.e. the family [itex](Df(x))_{x \in A}[/itex] is uniformly lipschitz of constant M on R^n). Then for any x_1, x_2 in A, we have ||f(x_2)-f(x_1)||<=M||x_2-x_1||." If m=1, then this is just the mean value theorem (MVT) plus the triangle inequality. But otherwise, the MVT applied to each component of f separately only leads ||f(x_2)-f(x_1)||<=mM||x_2-x_1||. So the proof suggested by the book I'm reading is that we write f(x_2)-f(x_1) using the fondamental theorem of calculus (FTC) as [tex]f(x_2)-f(x_1)=\int_0^1\frac{d}{dt}f(x_1+t(x_2-x_1))dt=\int_0^1Df(x_1+t(x_2-x_1))(x_2-x_1)dt[/tex] and then use the triangle inequality for integrals to get the result. But notice that the integrand is an element of R^m. So by the above, they certainly mean [tex]f(x_2)-f(x_1)=\sum_{j=1}^me_j\int_0^1Df_j(x_1+t(x_2-x_1))(x_2-x_1)dt[/tex] which does not, to my knowledge, allows for a better conclusion than ||f(x_2)-f(x_1)||<=mM||x_2-x_1||. Am I mistaken? Thanks!