Reference for a bilinear form lemma

I encountered the result below in a paper of Claude Viterbo (Intersection de sous-variétés lagrangiennes, fonctionnelles d’action et indice des systèmes
hamiltoniens, p. 379) that I was reading, and it does not have a reference. If anyone could provide me a reference, it would be very helpful.

Lemma: Let $Q^t$ be a $C^1$ family of bilinear forms defined in a Hilbert space. Let $Q^t$ be nondegenerate for $tneq 0$ and $Q^t = U_t + C_t$, where $U_t$ is positive definite with continuous inverse and $C_t$ is compact.
If $left.frac{d}{dt}Q^tright|_{t=0}$ when restricted to $ker(Q^0)$ has signature $sigma$ and nullity $mu$ we have
begin{equation}
sigma – mu leq index(Q^{-1}) – index(Q^{+1}) leq sigma + mu.
end{equation}

In fact, I am trying to understand the proof of proposition 8.

Thank you very much!