I've been trying to solve this topology problem but seem to have hit a snag. Any advice would be appreciated.
Let X = $C(\mathbb{R},\mathbb{R})$ Consider the family $(f_n)\subset X$ given by $f_n(x) = \sin(\frac{7n^2}{2n^2+1}x)+\frac{5n^3}{n^3+1}x$
Show that this family converges uniformly to some f on compact subsets of $\mathbb{R}$ but not uniformly to f on $\matbb{r]$
I have said that $f(x) = sin(3.5x) + 5x$ and that the supremum of the distance between $f_n(x) and f(x)$ is equal to
$|\sin(\frac{7n^2}{2n^2+1}x)-sin(3.5x)-\frac{5x}{n^3+1}|$
From there I have used the Heine Borel property to state that this supremum is between 0 and, for the compact K written as a subset of a closed interval between $[-C,C]$
|$\lim{5C/n^3+1}-\lim{\sin(\frac{7n^2}{2n^2+1}x)-\sin{3.5}}$|
Since Both limits tend to 0, by the squeeze theorem this is 0 and f_n(x) is uniformly convergent in compact subsets.
However, I'm unsure how to prove that it is uniformly convergent generally, and feel like the answer I have already written is incorrect towards the end.
Thank you.