Problem: Show that the sequence $$\gamma_n(t)= \left(\frac {1}{1+nt},\frac {t}{n}\right)$$ does not converge uniformly on $[0,1]$.
Attempt: Suppose $ε = \dfrac {1}{2}$. Then $$\lim_{n \to \infty} \gamma_n(t) = (0,0).$$ Then $\|\gamma_n(t)-\gamma(t)\| \ge ε$ yields $$\left\|\left(\frac {1}{1+nt},\frac {t}{n}\right)\right\| \ge ε.$$
Thus \begin{align*} \left\|\left(\frac {1}{1+nt},\frac {t}{n}\right)\right\| &= \sqrt{\left(\frac{1}{1+nt}\right)^2+\left(\frac{t}{n}\right)^2}\\ &= \frac {\sqrt{n^2t^4+2nt^3+t^2+n^2}}{n(nt+1)} \ge ε. \end{align*}
This is as far as I've gone with the solution since I'm not sure where to go. I know I need to find an $N \in \mathbb R$ that makes this inequality true. I'm quite sure what trick to use to minimize the inequality to find $N$.
Hints and advice would be helpful.
Consider $||\gamma _n(1/n)-(0,0)||$.