Let $X_i$ be independent, mean zero, $n\times n$, symmetric random matrices. $\|X_i\|\leq K$ almost sure for $\forall I$.

We have matrix Bernstein's inequality for the tail probability as follows

$$\forall t\geq 0,P(\|\sum X_i\|\geq t)\leq 2n \cdot\text{exp}(-\frac{t^2}{\sigma^2+Kt/3})$$ where $\sigma^2=\|\sum \mathbb{E}X_i^2\|$.

My question is how to prove $$\|\sum X_i\|\lesssim\|\sum \mathbb{E}X_i^2\|^{1/2}\sqrt{1+\log n}+K(1+\log n)?$$