The puppy is indexed by . thus that is what you're looking to make a statement about. everything in this problem centers around what you're looking at. Now by choosing an , you're taking a step toward putting a condition on . Without putting a condition on , how can you prove convergence when the sequence of functions only have meaning in relation to . So you restrict the statement to those functions that have . By doing so, you guarantee that . In those cases, converges to 0.