Help with Resnick's Probability Path (Lemma 1.3.1)

[pearsond88]

I am working through the proof of Lemma 1.3.1(a)
on page 7 of my edition.
The equations are too complex to try to type here,
so I hope that readers will kindly open their own
copies to follow what I am saying.

The proof starts "If", then there is a display equation
that I understand. Then one-and-a-bit lines of
text and inline equations and then a display equation
with the following structure:

Big_Sigma greater_or_equal Big_Sigma = infinity

This says that the series in the first Big_Sigma
has a greater sum than that in the second Big_Sigma.
This is because all the terms are zero or one,
and the second Big_Sigma is a sub-series of the
first, so the first must add up to more than
the second. Then, the second is infinite, therefore
the first is infinite also.

This would follow if the subseries (second Big_Sigma)
could not contain repeated terms from the first.
But it seems to me that it can contain repeated
terms, for example as follows:

n k_n

1 2
2 2
3 4
4 4
5 6
6 6
7 8
8 8
9 10
10 10
etc... etc...

Then if some omega were in A_p (for even p) but not
A_q (for odd q) then
the first Big_Sigma partial sum (to j=10) would
be 5, and the second would be 10. Thus the
inequality for the partial sum would be violated.
This can be extended indefinitely, violating
the inequality in the limit also. It is easy to
invent other simple examples.

Now the first Big_Sigma would still be infinite,
but not for the reason that I think Resnick
means. So the proof is flawed.

More likely is that I am wrong, and I would be
grateful for any guidance on this.

 

[mighty2000]

Thank you for bringing up this issue. As a fellow scientist, I understand the frustration of trying to work through complex proofs. However, I believe there may be a misunderstanding in your interpretation of the proof.

The key point of the proof is that the series in the first Big_Sigma is a subseries of the series in the second Big_Sigma. This means that every term in the first series is also included in the second series. In your example, the series in the first Big_Sigma would be 2+4+6+8+10+... and the series in the second Big_Sigma would be 2+2+4+4+6+6+8+8+10+10+... Notice how every term in the first series is also included in the second series.

The proof is not saying that the second series cannot contain repeated terms from the first series. In fact, it acknowledges that the second series may contain repeated terms. This is why the proof states that the first series must have a greater sum than the second series. In your example, the first series has a sum of 30 while the second series has a sum of 40. This still follows the inequality of the partial sums, as the sum of the first series is still greater than the sum of the second series.

I understand your concern about the proof being flawed, but I would suggest carefully reviewing the proof again and considering the concept of subseries. I would also recommend seeking guidance from your peers or the author of the proof if you are still unsure. Science is all about questioning and seeking clarification, so do not hesitate to ask for help when needed. I wish you the best in your understanding of Lemma 1.3.1(a) and in your scientific endeavors.