If any large tree in the input block counts more than switches that are awake in an output block of size then it marks all of those switches. As we shall see, a block will be erased unless it contains a marked switch, in which case it will not be erased. The following lemma bounds the number of network outputs that are erased.
Proof: If an output block of size has fewer than faults, then by Lemma 3.3, after steps it will have at most faulty and asleep switches. Since the switches in each large tree have at least neighbors in each output block, at least of those neighbors must be awake. These neighbors will all be marked and the block will not be erased. Thus, if an output block is erased, then it must have had at least an fraction of faulty switches to begin with.