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?
Best Answer
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.