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Let . Then there are only a finite number of string in with length less than or equal to n. This clearly imlies that every derivation of w eventually halts, enters a cycle or generates a sentential form of length greater than n.

when the rule that has been nondeterministically selected cannot be applied to the current sentential form, then the computation halts at step 3. Cyclic derivations are terminated in step 5(ii). If the length of a sentential form derived at any step exceeds the length of the previous sentemtial form, then the derivation is terminated in step 5(iii).

For any string ,, there is a derivation Sw without any cycle. Such a derivation can be simulated by the nondeterministic TM M using the above sequence of steps to accept w. Since every computation of M halts, it can be considered as a memebership testing algorithm for L(G) and hence L(G) is recursive.

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