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My previous/alt account is yetAnotherUser@feddit.de which will be abandoned soon.

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  • Switzerland has:

    • the lowest amount of area per km of track, except for micro nations
    • a fairly low amount of population per km of track - among the top 10 if population density is considered
    • lost less than 10% of tracks since its historical peak
    • a majority nationalized rail network
    • (as you mentioned) a fully electrified network

    While I haven’t travelled in Japan by rail (or any other mode), I have been to Switzerland. From what I’ve heard, in Japan there are many smaller local stations, where an ancient train arrives a few times a day.

    Whereas in Switzerland, it seems like nearly every local station has at least one train per hour.



  • Yes, but similar flaws exist for your proof.

    The algebraic proof that 0.999… = 1 must first prove why you can assign 0.999… to x.

    My “proof” abuses algebraic notation like this - you cannot assign infinity to a variable. After that, regular algebraic rules become meaningless.

    The proper proof would use the definition that the value of a limit approaching another value is exactly that value. For any epsilon > 0, 0.999… will be within the epsilon environment of 1 (= the interval 1 ± epsilon), therefore 0.999… is 1.



  • Unfortunately not an ideal proof.

    It makes certain assumptions:

    1. That a number 0.999… exists and is well-defined
    2. That multiplication and subtraction for this number work as expected

    Similarly, I could prove that the number which consists of infinite 9’s to the left of the decimal is equal to -1:

    ...999 = x
    ...990 = 10x
    
    Calculate x - 10x:
    
    x - 10x = ...999 - ...990
    -9x = 9
    x = -1
    

    And while this is true for 10-adic numbers, it is certainly not true for the real numbers.