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Blast into Math! Analatic nummer theora: ants, ghosts and giants

10. What are some of today’s unsolved number theory problems? Investigate and explore
current number theory research.

7.7 Examples and hints

There are few things more disappointing in mathematics than giving up on a problem too early only to
realize after looking at the hint that you could have solved it on your own had you persisted. So, give
yourself the chance to succeed independently, without any hints: believe in yourself! Take some extra
time to work on these problems. If you struggle with some of the problems, re-read some or all of the
chapter before looking at the hints. You could surprise yourself and solve something which you thought

was impossible. There are few things more satisfying than working hard to eventually solve a problem
which you thought was impossible. Because these are the last exercises in this book, don’t spoil your
mathematical fun by looking at the examples and hints too early.

• Hint for #1: Try a proof by contradiction, so assume that L is not non-negative, which
means that
L< 0.

Think of the sequence as ants on the number line, and their goal is to reach L. Where are the
ants on the number line? The terms in the sequence are all positive, so where are the ants? Is
it possible for them to get to L ? With a clever choice of in the definition of limit, you can
prove that it’s impossible for the definition of limit to be satisfied in case L< 0. This proves
that it must be true that L ≥ 0. Notice that it is possible for a sequence of positive numbers
to converge to 0, because we proved that

1
lim =0.
n→∞ n

• Hint for #2: You can prove this using only the definition of limit of a sequence.
• Hint for #3: If you can prove that the sequence is monotone, then you can apply the
Marching Ant Proposition to prove that it converges. Once you have proven that the
sequence converges, apply #2 with k =1.

• Hint for # 4: The series converges, so let’s call the limit L . Then, using only the definition
of limit, we know that for any ghost number > 0, there exists a giant number N ∈ N
such that
N

∀n ∈ N, n >N, x m − L <.

m=1

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