For a continuous-time signal $x(t)$ that is bandlimited (in the baseband) to $[-W,W]$, the standard proof of Nyquist sampling theorem proceeds in the frequency domain by examining the Fourier transform $X_s(f)$ of the $x(t)$ sampled at rate $2W=1/T$, and then showing that one can re-construct $x(t)$ using the low-pass filter with $\operatorname{sinc}(t/T)$ as impulse response to input of samples modulated by a Dirac comb. The standard proof uses the Fourier transform $X(f)$ of the original signal $x(t)$, which is computed over an infinite time domain, thus implicitly using infinite number of samples.
I am wondering if there is an alternative proof of the sampling theorem that, given the same conditions on the signal $x(t)$ as in the standard proof (band-limited, continuous, etc) uses fixed number of samples $n$ (as opposed to infinite number in the standard version), and then proceeds to show that, if sampled at rate $2W$, the reconstruction of the signal $\hat{x}(t)$ gets "closer" to original signal $x(t)$ as one increases $n$ such that $\lim_{n\rightarrow\infty}\|x(t)-\hat{x}(t)\|=0$
for some metric $\|\cdot\|$ like $L_1$ or $L_2$.
I don't have the mathematical sophistication to prove this myself, but I feel that this might be in the literature somewhere... Perhaps this an extension of the standard proof somehow.