Let $c: [a, b] \to M$ be a geodesic on a Riemannian manifold $(M, \left< \cdot, \cdot \right>)$. Define $\mathcal{V}_c := \Gamma(c^* TM)$, the space of vector fields along $c$, and $\hat{\mathcal{V}}_c$ the vector fields $V \in \hat{\mathcal{V}}_c$ satisfying $V(a) = V(b) = 0$. Consider the norm on $\hat{\mathcal{V}}_c$ given by $||X||^2 = \int_a^b \left[\left<D_t X, D_t X\right> + \left<X, X\right> \right]dt$, and let $\hat{H}_c^1$ be the completion of $\hat{\mathcal{V}}_c$ w.r.t. $||\cdot||$. This space is easily identified with the Sobolev space $H^1_0(I, \mathbb{R}^n)$.
We define the index form $I$ as a symmetric bilinear form on $\hat{H}_c^1$ given by $I(X, Y) = \int_a^b \left[\left<D_t X, D_t Y\right> - \left<R(\dot{c}, X)Y, \dot{c}\right>\right]dt$. Define the index of $c$, Ind$(c)$, as the dimension of the largest subspace of $\hat{H}_c^1$ on which $I$ is negative definite.
Lemma 4.3.2 of Riemannian Geometry and Global Analysis by Jost states that Ind$(c)$ is finite. To prove this statement, Jost supposes that it does not hold, so that we may assume the existence of a sequence $(X_n)\subset \hat{H}_c^1$ such that $I(X_n, X_n) \le 0$ and such that $(X_n)$ is orthonormal w.r.t. the $L^2_c$ product.
From $I(X_n, X_n) \le 0$, we immediately obtain $\int_a^b \left< D_t X_n, D_t X_n \right>dt \le \int_a^b \left<R(\dot{c}, X_n)X_n, \dot{c}\right>dt$.
Jost then concludes that $\int_a^b \left<R(\dot{c}, X_n)X_n, \dot{c}\right>dt \le E(c)\sup|R|$, where $E$ is the energy of $c$ and $R$ is the curvature tensor on $M$. Why does this hold? What does $\sup |R|$ actually mean here? Is it related to the extension of the Frobenius norm discussed here?
My first thought was to consider the vector field $Z(t) = \frac{\dot{c}(t)}{\sqrt{E(c)}}$, so that $Z$ is smooth and $||Z||_{L_c^2} = 1$. We then obtain $\int_a^b \left<R(\dot{c}, X)Y, \dot{c}\right>dt = E(c) \int_a^b \left<R(Z, X_n)X_n, Z\right>dt$.
Fix $p\in M$. Then $R_p\colon T_pM \times T_pM \times T_pM \to T_pM$ is multilinear. $\left(T_pM,\langle \cdot,\cdot\rangle_p \right)$ is Euclidean: there is an induced norm on the set $L\left(T_pM, T_pM, T_pM; T_pM\right)$, which is in our case $$ \left|R_p \right| = \sup_{u,v,w \in T_pM \text{ unit vectors}} \left\| R_p(u,v)w\right\| $$ (this is a particular case of the norm of continuous multilinear maps $E_1\times \cdots \times E_k \to F$ where all vector spaces are normed) and it has the property that $$ \left|\left\langle R_p(x,y)z,t\right\rangle_p \right| \leqslant \left| R_p\right|\|x\|\|y\|\|z\|\|t\| $$ by Cauchy-Schwarz inequality. Apply this to $p=c(t)$, $x=t=c'(t)$ and $y=z=X_n$. Also, note that $[a,b]$ is compact, so that $\sup_{t\in [a,b]} |R_{c(t)}| < + \infty$. What you want to show follows from this and from the definition of $E(c)$.