Martin-Löf randomness implies computable randomness

A successful martingale has $d(\epsilon )>0$, since a nonnegative martingale starting at $0$ is identically $0$. For each $n$, enumerate every string $\sigma$ with $d(\sigma )>2^{n}d(\epsilon )$ and let $U_n$ be the union of its cylinders. Strict comparison with a computable real is computably enumerable, so $U_n$ is effectively open.

The prefix-minimal threshold-crossing strings have the same union as $U_n$. The martingale conservation equation Computable martingales on binary strings bounds their total measure by $2^{-n}$ (their total capital cannot exceed initial capital), so $(U_n)$ is a Martin-Löf test.

Success puts the sequence in every level of that test, contradicting Martin-Löf randomness as defined in Martin-Löf tests and random sequences.