Let $R \in \mathbb{R}$ with $R>1$. Investigate the sequence of functions $ (f_n)_{n \in \mathbb{N}}$ with $$ f_n(x)= \frac{x^{2n}}{1+x^{2n}}, x\in [R, \infty)$$ with regard to uniform convergence.
I know the definition of pointwise and uniform convergence but never seen an example how to investigate uniform convergence, therefore I have no clue how to approach such problems. I know that pointwise convergence is necessary for uniform convergence but that's it. Can someone show me how to tackle such problems?
Necessary and sufficient condition for uniform convergence on set $A$ is $$\lim\limits_{n\to\infty}\sup_{x \in A}|f_n(x)-f(x)|=0$$ As in your case $A=[R, \infty)$ with $R \gt 1$, then $f(x)=1 $ and $|f_n(x)-f(x)|=\left|\frac{x^{2n}}{1+x^{2n}}-1 \right|=\frac{1}{1+x^{2n}}$. Now having supremum $\frac{1}{1+R^{2n}}$ we can obtain uniform convergence.