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.