Why is the inductive effect of σ-electrons only along saturated carbon chains?

In my book the following is written:



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> When an electron withdrawing atom such as a halogen is attached to the end of a carbon chain, the $\sigma$-electrons of the $\ce{C-X}$ bond are attached to or displaced towards the more electronegative halogen atom, as a result $\ce{X}$ acquires a small negative charge and $\ce{C\_1}$ acquires a small positive charge. The small positive charge on the $\ce{C\_1}$ in turn attracts the $\sigma$-electrons of the $\ce{C\_1-C\_2}$ bond towards it. As a result $\ce{C\_2}$ acquires a small charge, of course smaller than on $\ce{C\_1}$. 
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Then as $\ce{C\_2}$ acquires a small a charge shouldn't the positive charge of $\ce{C\_1}$ become less than earlier? Shouldn't it be true?



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> ...This type of a displacement of $\sigma$-electrons along a saturated carbon chain whenever an electron withdrawing group is present the end of the chain is called inductive effect.
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Why is there displacement of only $\sigma$-electrons not $\pi$-electrons and why does it happen along a saturated carbon chain only?


