(related to Corollary: Limit of N-th Roots)
Note that the \(n\)-th roots can be identified with the exponential function of some positive real base \(a > 0\): \[\sqrt[n]a=\exp_a\left(\frac 1n\right).\]
Also note that the sequence \(\left(\frac 1n\right)_{n\in\mathbb N}\) is a convergent real sequence with \(\lim_{n\to\infty}\frac 1n=0\). Because the exponential function of general base is continuous, it follows from the definition of continuity, the definition of the exponential function of general base and the corresponding proposition that
\[\lim_{n\to\infty} \sqrt[n]a=\lim_{n\to\infty} \exp_a\left(\frac 1n\right)=\exp_a(0)=\exp(0\cdot \ln(a))=\exp(0)=1.\]
