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Re: on reading ``o(1)''

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Daniel Bastos <dbastos@toledo.com> writes:

> Ben Bacarisse <ben.usenet@bsb.me.uk> writes:
>
>> Daniel Bastos <dbastos@toledo.com> writes:
>>
>>> Suppose f in o(1).  By definition, there's a natural number n(0) such
>>> that for any c > 0
>>>
>>>   0 <= f < c * 1 = c
>>>
>>> for any n >= n(0).  (AFAIK, c is a real number.)  That seems to imply
>>> that f = 0 because that's the only thing that could be less than any 
>>> c > 0.
>>
>> Not exactly.  First, I would not write n(0) so it looks like a function.
>
> This notation is great!  Let me defend it.  Consider n as a function
> that takes an argument 0 and gives you the constant ``n0'' --- as you
> like to write.  Functions can give anything you prefers to read. :-)

I respectfully disagree.  By all means make n a function but make it a
function of something useful.  As you use the notation, it is a
single-point function.  What are n(1), n(-e)?  What value does making it
a function have when there is only one point in the co-domain?

Secondly (and this is more important) you use n as not a function at the
very same time:  "for any n >= n(0)".  Now it's possible to unravel
that but, formally, the quantifier binds n in both locations.  At the
very least, use two names:  "there is an m(0) such that for all n >=
m(0) ...".

And to return to the first point, if you want to make it a function,
make it a useful one.  The n0 in question depends on c so make it a
function of c:

  For any c > 0 there is a natural number n(c) such that, for all k >=
  n(c), 0 <= f(k) < ck.

<snip>
> I picked this notation up with Brian M. Scott, an admirable
> mathematician who used to hang out at alt.algebra.help.

I strongly suspect he'd agree on this.  Making things into functions is
a great idea (and too few mathematician do it, relying instead on
subscripts and context to make it clear that this or that quantity is
linked to some other) but n is not a function when writing "for all n >
..." whatever goes in the ...

<snip>
-- 
Ben.

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on reading ``o(1)'' Daniel Bastos <dbastos@toledo.com> - 2017-03-24 21:29 -0300
  Re: on reading ``o(1)'' Kaz Kylheku <336-986-7787@kylheku.com> - 2017-03-25 02:40 +0000
  Re: on reading ``o(1)'' Michael Forsythe <forsythe@example.com> - 2017-03-25 05:51 +0000
    Re: on reading ``o(1)'' Ben Bacarisse <ben.usenet@bsb.me.uk> - 2017-03-26 22:14 +0100
      Re: on reading ``o(1)'' Michael Forsythe <forsythe@example.com> - 2017-03-28 10:32 +0000
      Re: on reading ``o(1)'' Michael Forsythe <forsythe@example.com> - 2017-03-28 10:33 +0000
  Re: on reading ``o(1)'' Ben Bacarisse <ben.usenet@bsb.me.uk> - 2017-03-26 22:11 +0100
    Re: on reading ``o(1)'' Daniel Bastos <dbastos@toledo.com> - 2017-03-28 21:05 -0300
      Re: on reading ``o(1)'' Ben Bacarisse <ben.usenet@bsb.me.uk> - 2017-03-29 14:23 +0100
        Re: on reading ``o(1)'' Daniel Bastos <dbastos@toledo.com> - 2017-04-03 09:40 -0300

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