How does existence of alpha and beta form of glucose prove that it exists as a cyclic structure
According to Wikipedia:


  Glucose is usually present in solid form as a monohydrate with a closed pyran ring (dextrose hydrate). In aqueous solution, on the other hand, it is an open-chain to a small extent and is present predominantly as α- or β-pyranose, which partially mutually merge by mutarotation.




Glucose predominantly occurs in nature in the form of the D‐enantiomer, which is generally believed to exist in three crystalline forms: $\alpha$‐D‐glucose monohydrate (Figure 1A)(Ref.2), anhydrous $\alpha$‐D‐glucose (Figure 1B)(Ref.2), and anhydrous $\beta$‐D‐glucose (Figure 1C)(Ref.1,3). Both anhydrous $\alpha$‐D‐ and $\beta$‐D‐glucose crystals are orthorhombic while $\alpha$‐D‐glucose monohydrate crystals are monoclinic (see Fig. 1A-C). However, a fourth form, which is metastable in solution phase at $\pu{38\!-\! 50 ^{\circ}C}$ and thought to be a hydrated form of $\beta$‐D‐glucose, has been reported as well (Ref.1,4).

The crystal structure of $\beta$-D-glucose published in 1960 (Ref.5) clearly showed the exsistence of pyranose ring system. As in the inserted box in Figure 1 state that, in aqueous solutions, 99% of D‐glucose exists as a mixture of the $\alpha$- and $\beta$-forms (approximately 62%  $\beta$ and 38% $\alpha$ when equilibrated at $\pu{31 ^{\circ}C}$ (Ref.1). Recent NMR study using fully $\ce{^13C}$ labelled glucose (Ref.6) clearly showed $\alpha/\beta$ ratio of $37/63$, which is almost identical to this literature value (Figure 2):





References:


F. W. Schenck, "Glucose and Glucose-Containing Syrups," In Ullmann's Encyclopedia of Industrial Chemistry: Ullmann's Food and Feed, Vol. 2; B; Elvers, Ed.; Wiley‐VCH Verlag GmbH &amp; Co. KGaA: Weinheim, Germany, 2017, pp. 781-802 (https://doi.org/10.1002/14356007.a12_457.pub2).
W. B. Newkirk, “Manufacture and Uses of Refined Dextrose,” Industrial &amp; Engineering Chemistry 1924, 16(11), 1173-1175 (DOI: 10.1021/ie50179a028).
G. R. Dean, “Optical-Crystallographic Properties of $\beta$-D-Glucose,” Anal. Chem. 1973, 45(14), 2440–2441 (DOI: 10.1021/ac60336a005).
G. R. Dean, “An unstable crystalline phase in the D-glucose-water system,” Carbohydrate Research 1974, 34(2), 315–322 (https://doi.org/10.1016/S0008-6215(00)82906-7).
W. G. Ferrier, “The crystal structure of $\beta$-D-glucose,” Acta Cryst. 1960, 13, 678-679 (doi: 10.1107/S0365110X60001588).
T. Richter, S. Berger, “A NMR method to determine the anomeric specificity of glucose phosphorylation,” Bioorganic &amp; Medicinal Chemistry 2013, 21(10), 2710–2714 (https://doi.org/10.1016/j.bmc.2013.03.008).