 Literal-frequency greedy approximation for MAX-SAT

In the residual unsatisfied formula, choose a Boolean literal contained in the largest number $a$ of clauses and make it true. Its opposite occurs in at most $a$ clauses, since it too was an eligible literal. The step permanently satisfies $a$ clauses; only clauses containing the opposite can become empty, so it loses at most $a$ clauses. Delete satisfied clauses and false literals and repeat. No satisfied clause or newly empty clause is counted twice. Summing gives total lost nonempty clauses at most total satisfied clauses $G$. All initially nonempty clauses are eventually in one of these classes, so $G\ge N_{+}/2\ge \mathrm{OPT}/2$. Initial empty clauses can never be satisfied and are excluded from $N_{+}$. This proves the approximation algorithm guarantee even with tautological clauses or repeated literals.