How to show that $\lim_{n \to +\infty} n^{\frac{1}{n}} = 1$?

You can use $\text{AM} \ge \text{GM}$.

$$\frac{1 + 1 + \dots + 1 + \sqrt{n} + \sqrt{n}}{n} \ge n^{1/n} \ge 1$$

$$ 1 - \frac{2}{n} + \frac{2}{\sqrt{n}} \ge n^{1/n} \ge 1$$

Let $\epsilon>0$. Choose $N$ so that ${1\over N}<\epsilon$. Noting that ${ n+1 \over n}<1+\epsilon$ for $n\ge N$: $$ N+1\le N(1+\epsilon) $$ $$ N+2 \le (N+1)(1+\epsilon)\le N (1+\epsilon)^2 $$ $$ N+3 \le (N+2)(1+\epsilon)\le N (1+\epsilon)^3 $$ $$\vdots$$ $$\tag{1} N+k \le (N+k-1)(1+\epsilon) \le N(1+\epsilon)^k. $$ Using $(1)$, we have for $n\ge N$: $$ n=N+(n-N)\le (1+\epsilon)^{n-N}N; $$ which may be written as $$ n\le B (1+\epsilon)^n, $$ where $B=N/(1+\epsilon)^N$.

Thus, for $n\ge N$ we have $$\tag {2} \root n\of { n}\le B^{1/n}(1+\epsilon). $$ Since $\lim\limits_{n\rightarrow\infty} B^{1/n}=1$, it follows from $(2)$ that $\limsup\limits_{n\rightarrow\infty} \root n\of { n}\le 1+\epsilon$.

But, as $\epsilon$ was arbitrary, we must have $\limsup\limits_{n\rightarrow\infty} \root n\of {n}\le 1 $.

Since, obviously, $\liminf\limits_{n\rightarrow\infty} \root n\of {n}\ge 1 $, we have $\lim\limits_{n\rightarrow\infty} \root n\of {n}= 1 $, as desired.

One could also argue as follows:

Note $\root n\of n>1$ for $n>1$. For $n>1$, write $\root n\of n=1+c_n$ for some $c_n>0$. Then, by the Binonial Theorem we have, for $n>1$, $$\textstyle n=1 +nc_n+{1\over2} n(n-1)c_n^2+\cdots\ge 1+{1\over2}n(n-1)c_n^2; $$ whence $$ n-1\ge\textstyle {1\over2}n(n-1)c_n^2. $$ So, $c_n^2\le {2\over n}$ for $n>1$; whence $$ 0<\root n\of n -1=c_n\le \sqrt{2/n} $$ for $n>1$, and the result follows.