I've been told that X is a random variable with a discrete uniform distribution on the set (1/n,2/n,...,1) and have been asked to show that it converges in probability to U(0,1) which is a uniform distribution on the interval (0,1). However, I feel as if I'm missing information. I think I may need to find the pdf of X and us either the Delta Method or the Moment Generating Function Method to show convergence in distribution. Any help is greatly appreciated.

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    $\begingroup$ X does not have a density, though its limit in distribution does. You could try the limit of its Cumulative Distribution Function, or of its moment generating function or characteristic function $\endgroup$ – Henry Sep 22 '20 at 0:35
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    $\begingroup$ In this case, convergence of the CDF is especially straightforward $\endgroup$ – Thomas Lumley Sep 22 '20 at 1:36

As said in the comments, we can show that convergence of the CDF.

Let's call that discrete uniform variable $X_n$. For any $0\leq t\leq1$, we have:


Where $[tn]$ means the floor of $tn$. When $n\rightarrow\infty$, this expression converges to $t$. So, we have:


This means that the CDF of $X_n$ converges to the CDF of $\text{Unif}(0,1)$. This implies that $X_n\overset{\mathcal D}\rightarrow U(0,1)$ (convergence in distribution).


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