I am reading "Calculus on Manifolds" by Michael Spivak.
I cannot follow the logic of the proof of the above theorem.
Then each $v(U_i)$ is the sum of certain $t_j - t_{j-1}$. Moreover, each $[t_{j-1}, t_j]$ lies in at least one $U_i$ (namely, any one which contains an interior point of $[t_{j-1}, t_j]$), so $\sum_{i=1}^n v(U_i) \geq \sum_{j=1}^k (t_j - t_{j-1}) = b - a$.
I can understand that each $v(U_i)$ is the sum of certain $t_j - t_{j-1}$ and
I can understand that each $[t_{j-1}, t_j]$ lies in at least one $U_i$,
because $t_0, t_1, \cdots, t_k$ are all endpoints of all $U_i$.
But I cannot understand why $\sum_{i=1}^n v(U_i) \geq \sum_{j=1}^k (t_j - t_{j-1})$.
Please explain me.
That is because of the statement each $[t_{j-1}, t_j]$ lies in at least one $U_i$.
Let $w_j$ be the number of times $[t_{j-1}, t_j]$ is included in all those $U_i$, we know that $w_j \ge 1$ and $t_j -t_{j-1} > 0$.
$$\sum_{i=1}^nv(U_i)=\sum_{j=1}^kw_j(t_{j}-t_{j-1}) \ge \sum_{j=1}^k(t_{j}-t_{j-1})$$