My daughter has dual Citizenship at birth. Can she change that fact by giving up what she inherited at birth? No. She will always be born a dual citizen, like Ted Cruz.
A person can renounce a citizenship. In fact, it’s not uncommon when people apply for jobs requiring a security clearance. Then, a person may renounce, say, his English citizenship, and be a US citizen only. It happens all the time. Then, you’re no longer a dual citizen. I don’t, however, think that’s exactly germane to this discussion.