Synchronverters are grid-friendly inverters that mimic the operation of conventional synchronous generators. In this paper, the non-linear stability of the synchronverter in the sense of boundedness is investigated and proven for the first time. To this end, initially, the complete non-linear model of the synchronverter is obtained and then the field-excitation current loop of the synchronverter is implemented using a bounded integral control (BIC) structure, as proposed by the authors in the literature. While maintaining the original synchronverter operation near the rated value, the BIC additionally guarantees that the field-excitation current will remain bounded in a given range. Using non-linear analysis and input-to-state stability theory, it is proven that the frequency dynamics will remain bounded which leads to a bounded synchronverter voltage and finally results in the stability of the closed-loop system in the sense of boundedness. Extensive simulation results verify that the proposed synchronverter with the BIC approximates the behavior of the original synchronverter near the rated value with a guaranteed bounded performance.