How to prove energy is preserved in sampling using Parseval's Relation?
Since your signal is band-limited, it has a representation in the orthogonal system of shifted sinc functions,

$$x_c(t)=\sum_{n\in\Bbb Z} x_c(nT)\operatorname{sinc}\left(\tfrac tT-n\right)$$

where $\operatorname{sinc}(x)=\frac{\sin(\pi x)}{\pi x}=\int_{-1/2}^{1/2}e^{i2\pi fx}\,df$.

By the Plancherel identity of the continuous Fourier transform this leads to 
$$
\langle \operatorname{sinc}(x-m),\,\operatorname{sinc}(x-n)\rangle 
=
\int_{-1/2}^{1/2}e^{i2\pi f(n-m)}\,df
=
\delta_{m,n},
$$
 i.e., orthogonality of the unscaled functions. In the scaled variant above one gets, first by orthogonality and then by rescaling of the time parameter, the Parseval identity for the orthonormal basis $\tfrac1{\sqrt{T}}\operatorname{sinc}(\tfrac tT-n)$, $n\in\Bbb Z$, of the space of band-limited functions,

$$
\|x_c\|_2^2=\sum_{n\in\Bbb Z} |x_c(nT)|^2 \langle \operatorname{sinc}(\tfrac tT-n),\,\operatorname{sinc}(\tfrac tT-n)\rangle=T\,\sum_{n\in\Bbb Z} |x_c(nT)|^2
$$