I want to prove $P(\lim_n{\max_1^n X_i \over \log n}=1)=1$ where $X_i$ are i.i.d Exp$(1)$. I was able to to show that
$$\left\{\lim_n{\max_1^n X_i \over \log n}=1\right\}= \left(\lim\inf\left\{{\max_1^n X_i \over \log n}>1-\epsilon\right\}\right) \bigcap \left(\lim\inf\left\{{\max_1^n X_i \over \log n}<1+\epsilon\right\}\right)$$
$$=\lim\inf C_n\cap\lim\inf D_n\,\,\,\text{ (say)}$$
I was also able to show that $P({\lim\sup X_n \over \log n}=1)=1$ and $$\left\{{\lim\sup X_n \over \log n}=1\right\} = \left(\lim\sup\left\{{X_n \over \log n}>1-\epsilon\right\}\right) \bigcap \left(\lim\inf\left\{{X_n \over \log n}<1+\epsilon\right\}\right)$$ $$=\lim\sup A_n\cap\lim\inf B_n\,\,\,\text{ (say)}$$
I was further able to show $A_n\subset C_n\subset\cup_1^nA_i$ and $D_n\subset B_n$
How can I conclude the result I want?
PS: This question has been posted before but the answer use "Gumbel Distribution", I am looking for a more Set Theoretic / Borel Cantelli lemma type answer if possible
Here the question asked: Almost sure convergence of maximum of sequence of random variables