When a temperature is lower than a certain critical value, a superconducting sample undergoes a phase transition from a normal state to a superconducting state. This onset process of superconductivity can be studied as a Rayleigh quotient under the framework of the Ginzburg-Landau theory. In particular, I study the onset problem for a thin superconducting loop in a large magnetic field. This double limit problem was first carried out by Richardson and Rubinstein by using formal asymptotic expansions. I rigorously show that a one-dimensional Rayleigh quotient in the spirit of Gamma-convergence. The full Gamma-convergence of the Ginzburg-Landau functional for a thin domain and a large field is also obtained. The rigorous analysis in this thesis shows the validity of Richardson and Rubinstein's formal results. It is also shown that the Rayleigh quotient related to this onset problem has a periodic variation with a parabolic background. The parabolic background effect can be explained by a non-ignorable effect if finite-width cross-section of a thin superconducting sample. This illustrate the observation of the Little-Parks experiment.