I tried to rigorously prove the value of long-forward price at time t. The outline of the proof is given on slide 22 of "The introduction the derivatives" slides.

I am quite close to the correct answer except that my answer has the incorrect sign. I have attached my workings as a pdf. Proof.pdf (0,3 MB)

Any idea of where I went wrong?

If we borrow the PV of \(F_{0,T} - F_{t,T}\). then surely we would have to pay back \(F_{0,T} - F_{t,T}\). i.e. have a cashflow of -(\(F_{0,T} - F_{t,T}\))=\(F_{t,T} - F_{0,T}\)? which would then make our overall cashflow at T 2(\(F_{t,T} - F_{0,T}\))? which is then non-zero and makes the arbitrage argument not work?