Too Long; Didn't Read
It is possible to verify payments without running a full network node. A user only needs to keep a copy of the block headers of the longest proof-of-work chain. He can't check the transaction for himself, but by linking it to a place in the chain, he can see that a network node has accepted it. The verification is reliable as long as honest nodes control the network, but is more vulnerable if the network is overpowered by an attacker. Businesses that receive frequent payments will probably still want to run their own nodes.