There must be a proof in Cassels' Lectures on elliptic curves (Cambridge University Press, Cambridge, 1991).
Se also his masterly survey Diophantine equations with special reference to elliptic curves (J. London Math. Soc. 41 (1966) 193–291) and the historical essay Mordell's finite basis theorem revisited (Math. Proc. Cambridge Philos. Soc. 100 (1986), no. 1, 31–41).