Using the BFGS method, how can I prove that if $B_k$ is symmetric and positive definite, then $B_{k+1}$ is also symmetric and positive definite?

Question

The following problem came up in my class, and I'm having trouble solving it.

Knowing that the BFGS method is defined by

$$B_{k+1} = B_k - \frac{B_ks_ks_k^TB_k^T}{s_k^TB_ks_k } +\frac{ y_ky_k^T}{y_k^Ts_k }, $$ how can I prove that, if $B_k$ is symmetric and positive definite, then $B_{k+1}$ is also symmetric and positive definite?

There's a suggestion, that we could use the Wolfe conditions, along with the following result:

$\bullet$ If $B_k$ is symmetric and positive definite, then $\langle{x,y}\rangle=x^TB_ky$ is scalar product.

I tried using the definition of a symmetric matrix, to prove that $B_{k+1}$ is symmetric, but reached a dead end, and I can't seem to figure out how to solve this, using the suggestions we have.