Eigenvalues of a random symmetric matrix

Question

In this paper (http://www.math.tau.ac.il/~nogaa/PDFS/akv3.pdf), I am trying to make sense of two points in the proof of Claim 2 (on page 4).

1. How is $u^T Bu \geq \lambda_s(B)$?

I understand that $u^TAu = \lambda u^Tu$ and because $u$ is a unit vector and orthogonal, we have that $u^Tu=I$. Therefore, $u^TAu=\lambda$. But I don't understand how $u$ which is a unit vector in the eigenbasis of matrix A, can be connected in this manner to the $s$-th eigenvalue of B.

2. Later, why is $t \leq u^T(B-A)u$?

I couldn't put the lemma here and make it self-contained as the definitions are spread across the first few pages.