Consider a rectifiable curve of length $\Lambda(a,b)$ traced out by a vector $\bf r(t)$ as $t$ varies over an interval $[a,b]$. If $a<c<b$, let $C_1$ and $C_2$ be the curves traced out by $\bf r(t)$ as t varies over the intervals $[a,c]$ and $[c,b]$ respectively. Then $C_1$ and $C_2$ are rectifiable and, if $\Lambda (a,c)$ and $\Lambda (c,b)$ denote their respective lengths, we have $$\Lambda (a,b) = \Lambda (a,c) + \Lambda (c,b)$$.
In the proof of the above statement regarding the additivity of arc length, Apostol first proves that $C_1$ and $C_2$ are rectifiable, then that $\Lambda (a,c) + \Lambda (c,b) ≤ \Lambda (A,b)$, and finally he proves the reverse inequality.
For this last part, he states
We begin with any partition $P$ of $[a,b]$. If we adjoin the point $c$ to $P$, we obtain a partition $P_1$ of $[a,c]$ and a partition $P_2$ of $[c,b]$ such that $$|\pi(P)| ≤ |\pi(P_1)|+|\pi(P_2) $$
where $\pi(P)$ is an inscribed polygon. I do not understand why this is so? Since a rectifiable curve must be continuous, why would $|\pi(P)|$ ever be smaller than $|\pi(P_1)|+|\pi(P_2)$. Will they not always be equal?
Try to sketch an example.
Let your rectifiable curve be the upper half of a circle of radius 1 centered at the origin in the plane.
Let $[a, b] = [-1,1]$.
Let $P = -1, -0.5, 0, 0.5, 1$
Let $c = 0.3$
Then $P_1 = -1, -0.5, 0, 0.3$
And $P_2 = 0.3, 0.5, 1$
If you draw that, I think you'll understand the inequality.