I am reading Durrett's "Probability: Theory and Examples" section on the St. Petersburg paradox, and he concludes that the fair entry fee for playing N games of the St. Petersburg game is $\log{N}$ dollars per game, or $N\log{N}$ dollars in total. He comes to this conclusion by appealing to a modified version of the WLLN (Thm 2.26 in Durrett), and I have worked out an understanding of this theorem as well as its application to the St. Petersburg Paradox. However, I don't quite see how the use of the theorem shows that $\frac{S_n}{n}$ is approximated well by $\log{n}$ for large $n$. The relevant excerpt in the textbook is shown below and I would like to know more about the conclusion of the fair price. I've also attached some of my notes, which show that $\frac{S_n}{n}$ does not converge in probability to $\log_2{n}$. If this is false, then what is the significance of $\log_2{n}$ and what justifies it as a fair price for $n$ games? In my notes, I have shown that the "error" between $\frac{S_n}{n}$ and $\log_2{n}$ may vary by a difference that increases with $n$. Any help is appreciated.
