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With today's quantum processors venturing into regimes beyond the capabilities of classical devices [1-3], we face the challenge to verify that these devices perform as intended, even when we cannot check their results on classical computers [4,5]. In a recent breakthrough in computer science [6-8], a protocol was developed that allows the verification of the output of a computation performed by an untrusted quantum device based only on classical resources. Here, we follow these ideas, and demonstrate in a first, proof-of-principle experiment a verification protocol using only classical means on a small trapped-ion quantum processor. We contrast this to verification protocols, which require trust and detailed hardware knowledge, as in gate-level benchmarking [9], or additional quantum resources in case we do not have access to or trust in the device to be tested [5]. While our experimental demonstration uses a simplified version [10] of Mahadev's protocol [6] we demonstrate the necessary steps for verifying fully untrusted devices. A scaled-up version of our protocol will allow for classical verification, requiring no hardware access or detailed knowledge of the tested device. Its security relies on post-quantum secure trapdoor functions within an interactive proof [11]. The conceptually straightforward, but technologically challenging scaled-up version of the interactive proofs, considered here, can be used for a variety of additional tasks such as verifying quantum advantage [8], generating [12] and certifying quantum randomness [7], or composable remote state preparation [13].