Exercise 2.34 in the book A first course in Sobolev spaces - Leoni

Let $u:[a,b] \rightarrow \mathbb R^N$

i) Let $g:[a,b] \rightarrow \mathbb R^N$ and take one $x_0 \in [a,b]$. Prove that$$ \left\| \int_a^b g(x)dx\right\| \geq \int_a^b \|g(x) \| dx -2 \int_a^b \| g(x) -g(x_0) \| dx. $$

ii) Using part i), prove that if $u \in C^1([a,b], \mathbb R^n$), then

$$ \text{Var}_{[a,b]}u = \int_a^b \| u'(x) \| dx.$$

I am stuck at proving i). It seems to be not true when I take $N=1$, and suppose that $\int g^+> \int g^-$, the inequality now reads

$$ \int g^+ - \int g^- \geq \int g^++ \int g^- -\int_a^b | g(x) -g(x_0) | dx $$or I have to prove that

$$ \int_a^b | g(x) -g(x_0) | dx \geq \int_a^b g^-(x)dx,$$

which seems not true...

Also, suppose we have a true version of i), how do we proceed to prove ii) using i)?

Thank you for your help!