OIS Curve Construction (Short End) - Proofs?

[Bseic]

I'm currently struggling with some proofs for an assignment. I have constructed the required OIS curve but am unable to show the equations hold true.

OIS are defined for us as follows:

"In an overnight index swap, simple fixed interest is exchanged for floating interest compounded daily at the official cash rate. For maturities up to one year a single exchange of payments occurs at maturity, while for maturities greater than one year payments are exchanged annually with a short front stub if required. Fixed rate is set so that the sum of discounted fixed cash flows equals the sum of discounted floating cash flows, where official cash rates are projected from the so-called OIS curve. The rate quoted for an overnight index swap is the fixed rate."

The first equation we are required to show is:

The second equation we are required to show is:

The third equation we are required to show is:

The following quotes are provided:

While I understand these equations intuitively, I am unsure of how to formalize a proof.

Any assistance would be greatly appreciated.

Thank you.

 

[KnightImperial]

I am not able to answer the question, but could you please recommend some books or any other materials on curve construction? thanks