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Proving existence results in martingale theory using a subsequence principle

New proofs are given of the existence of the compensator (or dual predictable projection) of a locally integrable cádlág adapted process of finite variation and of the existence of the quadratic variation process for a cádlág local martingale. Both proofs apply a functional analytic subsequence principle. After presenting the proofs, we discuss their application in giving a simplified account of the construction of the stochastic integral of a locally bounded predictable process with respect to a semimartingale.