I am fighting with this problem:
$$Let\ A,B \in\;\mathbb{R^{nxn}}\ be\ symmetric.\ In\ addition\ \lambda_{amin}(A) \ge 2\ and\ \lambda_{amax}(B) \le 1.\ Show\ that\ the\ matrix\ A+B\ is\ invertible.\ $$
I should prove that and still I easily find matrices A and B that will make the argument not hold. $$i.e.\ A = \begin{bmatrix}5 & 7\\2 & 3\end{bmatrix} and\ B = \begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}$$ Then the $det(A+B)=0$ and, thus not invertible.
I should somehow use this theorem: $\lvert \delta A \rvert < 1/\lvert A^{-1}\rvert$
I cannot figure out why "A+B is invertible" would be true, if I can find matrices A and B that will prove otherwise?
Indeed, you're not using symmetric matrices in your example.
A proof for symmetric matrices.
If $A+B$ was not invertible, there would exist $u \neq 0$ such that $(A+B)(u) = 0$.
Hence $\Vert A(u) \Vert_2 = \Vert B(u) \Vert_2$. But that can't be as this leads to the contradiction: $$2 \Vert u \Vert_2 \le \lambda_{\textrm{amin}}(A) \Vert u \Vert_2 \le \Vert A(u) \Vert_2 = \Vert B(u) \Vert_2 \le \lambda_{\textrm{amax}}(B) \Vert u \Vert_2 \le 1\Vert u \Vert_2$$