Proof that length of a finite cover for a real interval $[a,b]$ is greater than $b-a$ whenever $b>a$ . Also do anyone agree that there is a mistake in the thir line of the proof? each $v(U_i)$ is the sum o certain $t_j-t_{j-k}$ where $k>1 $ or equal one

The intuition of this proof is very simple, and you should be able to understand it from the given text. For a more "formal" approach set $$\epsilon_{ij}:=\left\\{\eqalign{1\qquad&\bigl(\>[t_{j-1},t_j]\subset U_i\bigr)\>,\cr 0\qquad &({\rm otherwise})\ .\cr}\right.$$ As the $U_i$ form a cover of $[a,b]$ one has $\sum_i\epsilon_{ij}\geq1$ for all $j$. On the other hand $$\
u(U_i)=\sum_j\epsilon_{ij}(t_j-t_{j-1})\ .$$ It follows that $$\sum_i \
u(U_i)=\sum_{i,j}\epsilon_{ij}(t_j-t_{j-1})\geq \sum_j(t_j-t_{j-1})=b-a\ .$$