I have the following sequence: $$a_n = \int_1^2 \sqrt[n]{{e}^{x^2}}\, dx$$
My aim is to prove that ${\{a_n\}^\infty_2}$ converges.
My idea was simply showing that $\lim\limits_{n \to \infty} \int_1^2 \sqrt[n]{{e}^{x^2}}\, dx$ equals $\int_1^2 {e}^{{\frac{x^2}{\infty}}}\, dx$ and therefore equals $\int_1^2 {1}\, dx$ which equals $2-1$ (in other words, showing that the limit exists and is a finite number)
Is that enough to show that the sequence converges? or must i use epsilon defintion in order to do that?
Let $f_n(x) = e^{x^2/n}$ for $x \in [1,2]$. As OP observes, $f_n \xrightarrow[n \to \infty]{} 1$ pointwisely.
$f_n$ is monotone and continuous on a closed and bounded interval $[1,2]$, so by Dini's Theorem, we can upgrade the pointwise convergence to uniform convergence, which allows interchanging of integral and limit signs.
As a result, $a_n \xrightarrow[n \to \infty]{} \int_1^2 1 = 1$.