In what way does Monotone Convergence imply LPO?

I’m studying the constructive proof that Limited Principle of Omniscience is equivalent to the Bolzano-Weierstrass theorem. The last step in the original Mandelkern’s proof is showing that the monotone convergence theorem implies LPO, which is claimed to be immediate. But it seems to me that in order to arrive at LPO, one has to represent reals with their binary expansions to plug in monotone convergence (of reals). But the claim that a real has a binary expansion is known to be equivalent to LLPO, which is invalid in constructive mathematics. Is there a way to prove this without using LLPO?