We denote
$$p_{+} = \mathbb{P}(X' \in C_{+1}) \quad \text{and} \quad p_{-} = \mathbb{P}(X' \in C_{-1})$$
We will assume, without loss of generality, that $p_{+} \geqslant p_{-}$. We have shown the inequality
$$d(X, C)^{2} \leqslant 4\lambda^{2} + (1 - \lambda) d(X', C_{\varepsilon_{n}})^{2} + \lambda d(X', C_{-\varepsilon_{n}})^{2}$$
Show that for all $\lambda$ in $[0, 1]$
$$\mathbb{E}\left(\left.\exp\left(\frac{1}{8} d(X, C)^{2}\right)\right\rvert \, \varepsilon_{n} = -1\right) \leqslant \exp\left(\frac{\lambda^{2}}{2}\right) \mathbb{E}\left(\left(\exp\left(\frac{1}{8} d(X', C_{-1})^{2}\right)\right)^{1-\lambda} \cdot \left(\exp\left(\frac{1}{8} d(X', C_{+1})^{2}\right)\right)^{\lambda}\right)$$