A function $f$ is absolutely continuous on $[a,b]$ is defined by: for each $\varepsilon>0$, there is a $\delta>0$, for each finite disjoint open interval $\{(c_k,d_k)\}_{k=1}^n$ contained in $[a,b]$, we have $ \text{if}\,\, \sum_{k=1}^n (d_k-c_k)<\delta, \,\,\text{then}\,\, \sum_{k=1}^n\left|f(d_k)-f(c_k)\right|<\varepsilon. $
However, in the book I'm reading, it is said that there is a equivalent definition, say $ \text{if}\,\, \sum_{k=1}^n (d_k-c_k)<\delta,\,\, \text{then}\,\, \left|\sum_{k=1}^n f(d_k)-f(c_k)\right|<\varepsilon. $
However, I can not prove it. How?