Say $B$ is a real symmetric matrix of dimension $n$ and $A$ is another real symmetric matrix of the same dimension.  

- What needs to be the bounds on (which?) norm of $B$ to ensure that $\lambda_{max}(A+B) - \lambda_{max}(A) = O(\frac{1}{n} )$ ? 


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Intuitively it seems that one needs the $l_{\infty}$ norm of $B$ to be low compared to $A$. But I am looking for more quantitative statements to hit that $O(\frac{1}{n})$ thing.