={1}, L = {w : : w is the unary encoding of 10, n 1}. For example, 110 L; 1100 L; 11 L.

Part 1. Use the Real Language pumping theorem to show that L Real Languages. Start by setting w to 1, where i is the first power of ten k, or 100 if k < 100. Then show that for all possible values of y, pumping in y once would generate w' L.

Part 2. Start the proof the same way as in part 1, and explain how the proof would fail if you chose to pump out once instead of in once.