Dear Ms. Pratt,
Thank you for posting your request on History Hub!
We suggest that you submit a Name Index Search to the US Customs and Immigration Service (USCIS) to obtain a copy of your great grandmother’s records. If she did not complete the process, USCIS should be able to provide the certification on non-existence, and although your great grandmother won’t have a naturalization record, it is likely that INS did create a file that could be used to verify her citizenship status.
On 18 December 2014, Danish Parliament passed a bill to allow Danish citizens to become foreign nationals without losing their Danish citizenship, and to allow foreign nationals to acquire Danish citizenship without renouncing their prior citizenship. It went into effect on September 1, 2015. Prior to this, Denmark did not recognize dual citizenship. Please refer to Denmark: Dual Citizenship Recognized for more information.
Please review NARA’s Naturalization Records for more information.
We hope this is helpful. Best of luck with your family research!