Consider the sequence of functions $<f_n(t)>$, defined as $ f_n(t) = \begin{cases} e^{-t^2} & \text{if } -n \leq t \leq n \\ \frac{e^{-n^2}}{[1-n(t-n)]} & \text{if } n \leq t < n + \frac{1}{n} \\ \frac{e^{-n^2}}{[1+n(t+n)]} & \text{if } -n - \frac{1}{n} < t \leq -n \\ 0 & \text{if } |t| \geq n + \frac{1}{n} \end{cases} $
I know it is point-wise convenient to $e^{-t^2}$ over $\mathbb R.$ I cannot check its uniform convergent as $$|f_n(t)-e^{-t^2}|=\begin{cases} 0 & \text{if } -n \leq t \leq n \\ |\frac{e^{-n^2}}{[1-n(t-n)]}-e^{-t^2}| & \text{if } n \leq t < n + \frac{1}{n} \\ |\frac{e^{-n^2}}{[1+n(t+n)]}-e^{-t^2}|& \text{if } -n - \frac{1}{n}< t \leq -n \\ e^{-t^2}& \text{if } |t| \geq n + \frac{1}{n} \end{cases}$$ Now stuck. Please help.