Hulu and the iphone 5

I have a iphone 4 and watched Hulu thur my lcd tv and I have just got a iphone 5 and will have to replace my cables, the salepersom at radio shack told me I needed a Lighting to 30 pin adapter and to use my old 30 pin HDMI cables for my iphone 4. I just pluged it in today and all I get is sound no video on the my tv only but I do get video on the iphone 5 screen , I was told I really needed a Lightning Digital AV Adapter but was thinking is hulu still doing iphone to tv video output on the iphone 5?

Lightning Digital AV Adapter

Lightning Digital AV Adapter

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