It may be easier to see if you re-write condition 3 in Theorem 5.3 to say that if {pn} is a sequence of price vectors converging to a price vector p* where p* is not equal to 0 but p*k=0 for some k, then max{z1(pn},z2(pn),...,zn(p)} is unbounded above. (Strictly speaking this should be lim sup instead of max.)
The good for which excess demand becomes unbounded need not be the same good for which the price tends to zero.
In the example, choose {pn}=(1/n,1), say.