EAX is a mode of operation for block-ciphers to implement an authenticated encryption. The original paper of EAX proved that EAX is unforgeable up to O(2n=2) data with one verification query. However, this generally guarantees a rather weak bound for the unforgeability under multiple verification queries, i.e., only (2n=3) data is acceptable. This paper provides an improvement over the previous security proof, by showing that EAX is unforgeable up to O(2n=2) data with multiple verification queries. The authors' security proof is based on the techniques appeared in a paper of FSE 2013 by a researcher which studied the security of a variant of EAX called EAX-prime.