This document provides a proof of the sampling theorem through analysis in the time and frequency domains. It presents the key steps of ideal low-pass filtering to eliminate aliasing, downsampling to reduce the sample rate, upsampling by inserting zeros, and final filtering to recover the original signal from the reduced set of samples. Diagrams and MATLAB scripts are used to visualize the transformations in the frequency spectra at each step and demonstrate that the output of the system is equal to the input when it is ideally band-limited.