Consider the simple symmetric random walk on $\mathbb{Z}$. That is, let $X_1, X_2, \dots$ be i.i.d. random variables with
$$
P(X_i=1)=P(X_i=-1)=1/2,
$$ 
and define $S_n=X_1+\dots+X_n$ with $S_0=0$. As is well known, the sum $S_n$ is  (null) recurrent and satisfies the law of the iterated logarithm
$$
P\left(\limsup_{n\to \infty} \frac{S_n}{\sqrt{2n \log \log n}}=1\right)=1,
$$

Assume now that we restrict ourselves to the subset of realisations of $S_n$  where each realisation satisfies 

 1. $\frac{1}{n} S_n \to 0$.

 2. For any integer $m$, there are infinitely many values of $n$ such that $S_n=m$ or $S_n=-m$. 

Does the law of the iterated logarithm take a stronger form in this case, in the sense:

>>Q: If $S_n$ is satisfies the criteria above, does this imply
$$
\limsup_{n\to \infty} \frac{|S_n|}{\sqrt{2n \log \log n}}\leq 1?
$$

Or are there even in this case subsets of realisations for which this is not true?