PS4's resolution output to games

I want to get a PS4 for myself for Christmas but can't afford it and a new TV on top of that.

My question is, I have a Sharp 20 inch TV that displays at 1080i max, and I know most games now output at full 1080p, will games I get have to be output at the full 1080p? Or will they still display at 1080i?

They can output at 1080i, games support a range of resolutions

@Chris_53:Much appreciated sir.