Uniform integrability of $\cos ^n\left(\frac{x}{\sqrt{n}}\right) 1_{|x| \leq c \sqrt{n}}$

Question

I am trying to show uniform integrability of $\cos^n \left(\frac{x}{\sqrt{n}}\right) 1_{|x| \leq c \sqrt{n}}$, where $c$ is some positive constant. I was able to show that $\cos^n \left(\frac{x}{\sqrt{n}}\right) \to e^{-\frac{x^2}{2}}$ by using Taylor expansion of $\cos^n \left(\frac{x}{\sqrt{n}}\right) = e^{-\frac{x^2}{2}} e^{\frac{x^2}{2} + n \log{\cos\left(\frac{x}{\sqrt{n}}\right)}}.$ Could you help me with uniform integrability?

Answer 1

Each of these functions, in absolute value, is bounded by the constant function $1.$ Take $\delta = \epsilon.$