Uniform convergence relation to the limit of sup

For uniform convergence it is a requirement that $$\lim_{n\to\infty}\sup_{x\in \mathbb{R}}|f_n(x)-f(x)|= 0$$ so does showing this does not exist for the given function prove something does not converge uniformly for example $$f_n(x)=\frac{xn^{1/2}}{1+nx^2}$$ has limit function $0$ but $\underset{n\to\infty}{\lim}\underset{x\in \mathbb{R}}{\sup}|f_n(x)|$ does not exist do does this show that it is not uniformly convergent?

If $f_{n}\rightarrow f$ pointwise and $f_{n}\rightarrow g$ uniformly, then $f=g$ (check).
In your case, $f_{n}\rightarrow0$ pointwise. Moreover, you've shown that $f_{n}$ does not converge to $0$ uniformly. Therefore, $f_{n}$ does not converge uniformly (to any function).