[f97tosc]

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At some point you will will be 1 or 2 away from every number on the low side (i.e. before you pass or hit any number you will be exactly one or two less than it). For each of these two equally likely occurences

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I think you need to distinguish between on the one hand eventually being 1 or 2 below, and on the other hand reaching 1 below without having first reached 2 below (or vice versa).

For eventually reaching 1 or 2 below, then potentially both can happen, so I don't think that is what you mean.

On the other hand, if you mean reaching 1 below without having first reached 2 below, or vice versa, then it is not obvious, in fact I think it is false, that they are equally likely.