$T_n$ is a random variable having pdf $$f_n(t) = \begin{cases} n; & 0 < t < n^{-1}\\ 0; & \text{otherwise} \end{cases}.$$
Consider the the sequence $T_1, T_2, \cdots$ and give the pdf or the pmf of the limiting distribution.
Here is what I came up with: The support of $T_n$ is $(0,1/n)$, so we have that the CDF is $$F_{T_n} = \begin{cases} 1; & t \geq 1/n \\ nt; & 0 < t< 1/n \\ 0; & t \leq 0 \end{cases} $$
So when taking the limit as n goes to infinity, we have $$\lim_{n\to\infty} F_{T_n} = \begin{cases} 1; & t > 0 \\ 0; & t\leq 0 \end{cases}$$
Therefore, $T_n$ converges in distribution to a degenerate random variable T having pmf $$f_T(t) = I_{\{1\}}(t)$$
Can someone tell me if the limiting distribution that I have written above is correct (i.e. where I have the cases listed for the limit). I am having trouble determining whether those cases are correct because it seems that for any t greater than 0, the limit as n goes to infinity will be 1.