The Heine-Borel Theorem

Can't seem to make sense of the definition in my Calc book.  Time to lay it out.

Consider a finite number of points c_k on an interval I:
     
If I is an open interval (a,b), then  a < c₁,   and c_n < b.
If I is a closed interval [a,b], we may choose   c₁ = a    and/or    c₂ = b, but we do not have to.

Widder says:

6.1 THE HEINE-BOREL THEOREM
Let there correspond to each point c of the closed interval a ≤ x ≤ b a  number δ_c and an interval I_c of length 2δ_c with c the center point,

The Heine-Borel theorem states that a finite number of the intervals (1) can be chosen which will cover the whole interval a ≤ x ≤ b.  That is, every point of a ≤ x ≤ b will be in at least one of the above mentioned finite number of intervals (1).  In order to emphasize the need for proving this result, let us give an example to show it false if the interval (a, b) were open instead of closed.
    Let a = 0, b = 1, and define I_c, for 0 < c < 1, as

That is, δ_c = c/2.  No finite set of intervals (2) will cover the interval 0 < x < 1.  For, consider such a set, I_c1, I_c2, . . . , I_cn, where 0 < c₁ < c₂ . . . < c_n.  Of these, the interval I_c1  reaches farthest to the left.  Hence, no point to the left of c₁/2 is covered by the set.
                    − David Widder, Advanced Calculus; Dover Press,1989; pg 170

OBJECTIONS

1) But with δ_c= c/2, the closed interval [0, 1] cannot be covered by a finite number of intervals, either.  For consider a set   I_c1, I_c2, . . . , I_cn, where 0 = c₁ < c₂ . . . < c_n=1.  We cannot have c₁ = 0:
     
Which leaves us with:
    
And we cannot cover the range    0  ≤ x  <  c₁/2.

Allowing equalities at endpoints  does not ease the bind.  Let c₁ = 0.  Then we have
     
Now consider I_c2.  Its leftmost point is c₂/2.  We still cannot cover  0 < x <  c₂/2.

2) "Covered" remains undefined.

3) According to (1), each of the I_c has a nonzero length, 2δ_c.  If we allow the endpoints to be among the c_i, the "cover" can extend beyond the edges of the interval.  But if the cover may exceed the interval, we can cover an open interval as well: every point of (a, b) is covered by the same set that covers [a, b].
We may do this at any time by choosing a fixed width for δ_c.  Perhaps that is a trivial approach.  Perhaps Widder should specify what he expects of δ_c.

Note that, although we can approach 0 from above in such a way that we never arrive, division is bounded from below, and the limit exists:
       lim _{{n→ ∞}} 1/n  = 0.
 Enumeration is not bounded from above:
       lim_{{n → ∞}} n  = ∞.
{Oops, did I give away the mystery?}

If we do not allow ourselves to hold the limit point 0, and then ask how large we have to make n  to obtain zero, then of course there is no answer.

4) We can always use circular logic to prove our assumptions back to ourselves.
For suppose we cannot.
But this is contrary to our assumption, and so we can always use circular logic to prove our assumptions back to ourselves.

Widder measures differently when he has a closed, versus open, interval.  He uses offset (finite number of subintervals) when he has the limit point, and proceeds by infinite division when the limit cannot be used.
If we break his procedure up into distinct steps, he neglects to mention the steps in red for the closed interval:
    - subdivide any finite distance c − a
    - No finite number of divisions (intervals/inequalities) can obtain a.
    - By hypothesis, the limit exists (c − a is a finite quantity), so
    - Obtain the limit, a, by  infinite subdivision.
    - Is the limit a part of the interval?  
                           (yes)
    - agree to use it, instead of the limit sequence.
    - go back to ordinary measure and observe that we may lay off a finite number of subintervals between c and a.

The question in bold is where the two cases diverge.   In the middle of the argument, he asks if he can skip the proof altogether.  The answer is yes for the closed interval, no for the open.  Otherwise, his procedure is equally incapable of obtaining a closed interval in a finite number of steps.  The proof and definitions are irrelevant to the conclusion he obtains (see 5).

The problem remains: how ought they relate?

5) Suppose we choose our endpoint a arbitrarily small, but nonzero.  Leave δ_c,=c/2 as in the book example (2).  We get a different result.  The whole, open  interval a < x < b can be convered by a finite number of I_ck.  And likewise, a closed interval.
If the interval is open, let    a < c₁ < 3a/2.
  If the interval is closed,  we may use this same range for c₁.  In addition, we can let c₁=a, and the interval c₁/2 < x < 3c₁/2  is nonzero.

The procedure offered in (2) fails for any interval containing zero, where the width of the intervals I_ck  is some proper fraction of c.  Perhaps this is the result he wants.  He is about to use the Heine-Borel theorem to "prove" that a function which never escapes to infinity, over a finite range of x, has a *cough* upper limit.  
 But if, with a finite number of intervals of width 2δ_c,  where 2δ_c is a proper fraction of c, we can cover the entire range 0 ≤ x ≤ 1, then a finite number of intervals centered at c'_k = 1/c_k  and of width  δ'_c = 1/δ_c will cover the range 1 > x > ∞.  

6)  No interval containing 0 is compatible with both the definition (1) and the assumption of δ_c = c/2.  So there can be no subinterval centered at 0.  But in the following Theorem (see part 2), he requires that every point of the closed interval a ≤ x ≤ b  be the center of such a subinterval.  

7) We can obtain contrary results by choosing different functions for δ_c.  We have a situation where
Given ε there is a corresponding δ such that there are a finite number of ...
Given ε there is a corresponding δ such that there are not a finite number of...

Which is not a contradiction; δ_c is not unique or prescribed: we can measure in whatever fashion we wish. We may divide up a length by crossing half the distance between ourselves and the other side, and never arrive at the other side, as Xeno does, AND we can cross the whole distance in a single step.

Onward:                                                                                                   Part II: →