Dedekind's principle of continuity (which is equivalent to the completeness of the real numbers) appears in his construction of the real numbers, a construction that Kronecker did not accept.
Dedekind's principle of continuity (which is equivalent to the completeness of the real numbers) is the reason why these limits exist. However, this principle comes from Dedekind's construction of the real numbers, a construction that Kronecker did not accept.