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Suppose we have a sequence of random variables $(X_n)_{n \geq 1}$ and the law of each $X_n$ is supported by $\{1,2,...,n\}$. Hence, we can assume that they are all $\mathbb{N}$-valued random variables.

If we like to prove that the sequence $(X_n)_{n \geq 1}$ converges in distribution to the dirac measure at $\{1\}$, i.e. that $\mathbb{P}(X_n=1)\to 1$ as $n\to \infty$, would it be sufficient to show that $ \mathbb{P}(X_n=n-k) \to 0$ as $n\to \infty$ for $k\in \{0,1,...,n-2\}$ ? The growing support is what bothers me.

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Example: $\Bbb P(X_n=k)={1\over n}$ for $k=1,2,\ldots,n$. (That is, $X_n$ is uniformly distributed over $\{1,2,\ldots,n\}$.)

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