How can one prove that $\lim_{n \to \infty}a^{1/n}=1$ for every $a>0$?

Case 1: Suppose that $a\gt 1$. Let $a=1+\delta$, where $\delta$ is positive.

We claim that if $n \ge 1$ then $$1\le a^{1/n}\lt 1+\frac{\delta}{n}.\tag{1}$$ The fact that $\lim_{n\to\infty} a^{1/n}=1$ is an immediate consequence of Inequality (1).

To prove (1), suppose to the contrary that $a^{1/n}\gt 1+\frac{\delta}{n}$. Then $$a=(a^{1/n})^n\gt \left(1+\frac{\delta}{n}\right)^n.\tag{2}$$ But by the Bernoulli Inequality, $$\left(1+\frac{\delta}{n}\right)^n \ge 1+n\frac{\delta}{n}=1+\delta.\tag{3}$$ From (2) and (3) we conclude that $a\gt 1+\delta$, contradicting the fact that $a=1+\delta$.

Case 2: Suppose that $0\lt a\lt 1$. Let $b=\frac{1}{a}$. Then $b\gt 1$. By Case 1, $b^{1/n}$ has limit $1$, and therefore so does $a^{1/n}$.

Remarks: $1.$ The Bernoulli Inequality states that if $t\gt -1$, then $(1+t)^n\ge 1+nt$. We only need it for $t\gt 0$. In that case, it is an immediate consequence of the Binomial Theorem. There is also a quite simple induction proof.

$2.$ We gave a quite formal proof. But it comes down to the fact that a number $\gt 1$ raised to a large enough power is very large, and in particular greater than $a$.

Note: I first saw this in "What is Mathematics?" by Courant and Robbins.

You need two cases: $0 < a < 1$ and $a > 1$. These both use the motto "Always expand around zero."

These cases both use Bernoulli's inequality (if $x \ge 0$ and $n \ge 1$ then $(1+x)^n \ge 1+xn$). These also implicitly use the Archimedean axiom for the real numbers (for any positive reals $x$ and $y$ there is an integer $m$ such that $mx > y$).

If $a > 1$, let $a^{1/n} = 1+b$ where $b > 0$. Then $a =(1+b)^n \ge 1+bn $ so $b \le \frac{a-1}{n} $. We can make $b < \epsilon$ for any $\epsilon$ by choosing $n > \frac{a-1}{\epsilon} $.

If $0 < a < 1$, let $a^{1/n} = \frac1{1+b}$ where $b > 0$. Then $a = \frac1{(1+b)^n}$. As before, $(1+b)^n \ge 1+bn $, so

$\begin{align} &a \le \frac1{1+bn}\\ &\text{or}\\ &\frac1{a} \ge 1+bn\\ &\text{or}\\ &b \le \frac{1/a-1}{n}\\ &\text{or}\\ &1+b \le 1+\frac{1/a-1}{n} = \frac{n+1/a-1}{n}\\ &\text{or}\\ &\frac1{1+b} \ge \frac{n}{n+1/a-1} = 1-\frac{1/a-1}{n+1/a-1}\\ \end{align} $

We can make $\frac1{1+b} > 1-\epsilon $ by making $n+1/a-1 >\frac{1/a-1}{\epsilon} $ or $n >(\frac1{a}-1)(\frac1{\epsilon}-1) $ for any $\epsilon > 0$.

Check this and then use the squeeze theorem with:

$$(1)\;\;a>1\implies 1\le\sqrt[n] a\le\sqrt[n]n\;\;,\;\;\text{for}\;\;n\ge a\\(2)\;\;a<1\implies\;\text{use arithmetic of limits with}\;\;b:=\frac1a>1\ldots$$