Limit of $n^{\frac{1}{n}}$. We have to prove that $n^{\frac{1}{n}}$ converges to $1$. I have proved it using the binomial theorem where we can substitute $(1+t)$ in place of $n$ and proceed forward. However along with the question another approach was mentioned where we can make use of the Monotone Convergence Theorem. For this approach, I proved that $a_n=\left(1+\frac{1}{n}\right)^n$ is increasing and using $a_n<a_{n+1}$ I proved that $n^{\frac{1}{n}}$ is decreasing after the third term. Also it is bounded as all terms are greater than $0$. So by MCT it should be convergent. But I am not able to evaluate the limit and am only able to prove its existence here. Please help.

$$a_n = n^{1/n} \implies \ln a_n = \frac{\ln n}{n} \implies \lim\ln a_n = 0 \implies \ln \lim a_n = 0 \implies \lim a_n = e^0 = 1$$
