Nonexistence of Limit of Sum of Prime Factors

Question

In trying to prove the following problem, I find great difficulty in proceeding to generalizing some results:

Let $s(n)$ be the sum of prime factors of an integer $n$. Prove that $\lim_{n \to \infty} \left ( \frac{s(n)}{n} \right )$ does not exist by using the Subsequence Theorem.

While trying to understand this problem, I did the prime factorization of several integers: $n=2, s(n) = 2, \frac{s(n)}{n} = 1; n=3, s(n)=3, \frac{s(n)}{n}=1; n=4, s(n) = 2+2, \frac{s(n)}{n}=1; n=6, s(n)=2+3, \frac{s(n)}{n}= \frac{5}{6}; n=9, s(n)=3+3, \frac{s(n)}{n}= \frac{2}{3}; n=10, s(n)=5+2, \frac{s(n)}{n}= \frac{7}{10}$ etc.

As a naive guess, I would say to get a subsequence $\{b_n\}$ be the quotients that give us 1, i.e., when $n=2,3$, and 4. Then, I would get another subsequence of the quotients that approach 1, e.g., when $n= 6,9,10,$ and 15. This is exactly where I get confused and do not know how to proceed since I know that I cannot just write up a list of fractions.

Another approach I have thought of is separating the limit problem as follows: $\lim_{n \to \infty} \left ( \frac{1}{n} \right ) \cdot \lim_{n \to \infty} \left (s(n) \right )$ by using the limit laws. However, I don't know if this violates the initial condition of using the Subsequence Theorem. If it does not, then could I use the fact that the first limit diverges since it is the Harmonic series? Any help would be greatly appreciated, thanks in advance.

I am using the textbook Introduction to Analysis by Arthur Mattuck.

Answer 1

Hint: What is $\frac{s(n)}{n}$ if $n$ is a prime? And what if $n=2^k$?

Answer 2

In general you cannot separate a limit $\lim_{n \to \infty} a_n b_n$ into a product of limits to show that a limit does not exist, unless either $a_n \to a \neq 0$ or $b_n \to b \neq 0$. This does not hold for your particular choice of $a_n = s(n)$ and $b_n = 1/n$. The subsequence theorem you mentioned says that if a sequence converges to a limit then all subsequences must converge to the same limit. So go with another hint, and find two subsequences of $s(n)/n$ that converge to different limits; then your sequence doesn't converge.