I wasn't sure whether this question should be asked on Computer Science or Math Stack Exchange, so I will ask first here.
Having $X_i \sim N(0,I_n)$, s.t. $X_i \in \mathbb{R}^n$ and $i \in \{1,...,d\}$ and with high probability $\langle X_i, X_i \rangle = O(n)$, where every two $X_i,X_j$ are independent from each other, show that with high probability upper bound of the inner product is: $$|\langle X_i, X_j \rangle| \le O(\sqrt{n \log{d}})$$ where $ d \ge n$