I am working on some problems from an analysis book for self study to get a better idea of uniform convergence, and I am having a hard time visualizing what this function looks like, or really understanding if I am doing this correctly.
I am trying to show that the sequence of functions $$ \gamma_n(t) = \bigg(\frac{1}{1+nt}, \frac{t}{n}\bigg) $$
does not uniformly converge on [0,1]. The way I have done this is to show that the limit function (as $n\to\infty$) looks like this. $$ F(t) = \begin{cases} (1,0) \text{ if } t=0\\ (0,0) \text{ if } t \neq 0 \end{cases}. $$ Since the limit function is discontinuous, then this function cannot uniformly converge as there does not an exist an $n>N$ such that $$ ||\gamma_n(t) - F(t)|| < \epsilon \text{ for all } t\in [0,1]. $$
Is this the correct way about showing uniform convergence for these functions with multiple components? I feel like I don't quite understand what this type of function looks like, other than the final function is piecewise and therefore not continuous.