When you find a solution, use the :

For those seeking to master "Probability with Martingales," finding solutions to exercises and problems is an essential part of the learning process. The best approach often involves:

Most attempts just cite dominated convergence. This solution carefully constructs a subsequence argument and justifies uniform integrability without skipping steps.

Williams famously did not publish solutions – he believed in struggling productively. That’s great for a classroom, but for self-learners, getting stuck for days on Exercise 6.3 helps no one. A well-written solution guide becomes a learning tool , not a crutch, when you use it to check your reasoning after a genuine attempt.

Williams often hides hints or simplified versions of complex proofs in the back of the book. Essential Prerequisites