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In the problem, we are given a set of clauses, and we want to find an assignment that satisfies as many of them as possible.

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You can expand the brackets that will give you $m - \sum_{j=1}^{m}(\frac{1}{2})^{k_j}$. Now using the fact that $k_j \geq 1$ you can show that $(\frac{1}{2})^{k_j} \leq \frac{1}{2}$. Upper bound each term in the sum and you will obtain the desired inequality.

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what do you mean upper bound each term? –  skill sim Dec 3 '12 at 0:43
    
well, since each $(\frac{1}{2})^{k_j} \leq \frac{1}{2}$ then $\sum_{j=1}^{m}(\frac{1}{2})^{k_j} \leq \sum_{j=1}^{m}\frac{1}{2} \leq \frac{m}{2}$ –  Alexander Zimin Dec 3 '12 at 0:58
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