Proving inequality regarding arc length in Apostol's book

Question

If P={$t_0,t_1,...,t_m$} is a partition of [a,b] prove that the following inequality holds. $$\sum_{i=1}^{m}|f_k(t_i)-f_k(t_{i-1})|≤\sum_{i=1}^{m}||f(t_i)-f(t_{i-1})||≤\sum_{i=1}^{m}\sum_{j=1}^{n}|f_j(t_i)-f_j(t_{i-1})|$$

where $f=(f_1,...,f_n)$. I managed to prove the left side of the proof but the other side cannot be solved. This is from Apostol's book page 135. Would really appreciate some assistance.

Answer 1

You can get the inequality on the right simply by repeated application of the triangle inequality. The norm of a sum is less than or equal to the sum of the norms.

Answer 2

$\big(\sum_{j=1}^{n}|f_{j}(t_{i})-f_{j}(t_{i-1})|\big)^{2}$

$=\sum_{j=1}^{n}|f_{j}(t_{i})-f_{j-1}(t_{i-1})|^{2}+\sum_{j\neq k}|f_{j}(t_{i})-f_{j}(t_{i-1})||f_{k}(t_{i})-f_{k}(t_{i-1})|$

$\ge\sum_{j=1}^{n}|f_{j}(t_{i})-f_{j}(t_{i-1})|^{2}=\|f(t_{i})-f(t_{i-1})\|^{2}$

for each $i=1,...,m$.