Andrew Warren wrote:

>                     oo              m
>                    ---   9         ---   9
>         0.999... = >   ---- = lim  >   ----
>                    --- 10^n  m->oo --- 10^n
>                    n=1             n=1
>
>         Choose epsilon > 0. Suppose delta = 1/-log_10 epsilon, thus
>         epsilon = 10^(-1/delta). For every m>1/delta we have that
>
>         |  m           |
>         | ---   9      |     1          1
>         | >   ---- - 1 | = ---- < ------------ = epsilon
>         | --- 10^n     |   10^m   10^(1/delta)
>         | n=1          |
>
>         So by the (epsilon-delta) definition of the limit we have
>
>                m
>               ---   9
>          lim  >   ---- = 1
>         m->oo --- 10^n
>               n=1
>
> Does that make it clearer?


Hmmm. I feel like the child in the "emperors new
clothes". These experts all backing each other up re
what they agree is "right". Sort of like politicians
but with better formulas. ;o)

Surely any child could point out that with the
0.9999 issue that ANY POSSIBLE STEP can only reduce
the error to a smaller amount. The *number* of steps
is completely confusing and superfluous, the error
must always exist as there is NO POSSIBLE STEP
that can eliminate the error, only reduce it a bit
more. And you really don't need impressive formulas
to understand that reality? :o)
-Roman

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