Converted from [a comment](https://mathoverflow.net/questions/448349/perturbation-of-positive-semidefinite-matrix#comment1158466_448349):

> This is false: $$ A=\begin{pmatrix} 0 & 0\\ 0& 1\end{pmatrix},\quad B=\begin{pmatrix} 0 & 1\\ 1 & 0\end{pmatrix}$$ Then $A+\epsilon B$ is not positive definite for any $\epsilon\not= 0$.