Just because Naruto was ignored throughout most of his childhood,he never gave up on finding a friend. When he found out that Sasuke was only competing with him because he wanted to be able to be seen as the same level as himself,he knew it was only because he wanted to be strong to avenge his clan.
The reason he kept pushing him is because he wanted to make sure he could be pushed to near the edges of sanity and strength so he was strong enough to beat Itachi.
