Consider a sequence of random variables $\{X_n\}_n$. Given that
$$X_n - \sqrt{n} c \to N(0,1)$$ in distribution ($c$$c>0$ is a constant number bounded away from zero).
Show that for any constant $t>0$, we have
$$\lim_{n\to \infty} P(X_n>t)=1$$
[Update]
I use the following proof:
$$P(X_n>t)=P(X_n-\sqrt{n} c> t-\sqrt{n})=1-P(X_n-\sqrt{n}c<t-\sqrt{n}c)$$
$$\lim_{n\to\infty} P(X_n-\sqrt{n}c<t-\sqrt{n}c) = \lim_{n\to\infty} \Phi(t-\sqrt{n}c) = \Phi(-\infty)=0$$ where $\Phi$ is the CDF of $N(0,1)$.
So, we have $$\lim_{n\to\infty} P(X_n>t)=1$$
However, I am not sure whether the equation
$$\lim_{n\to\infty} P(X_n-\sqrt{n}c<t-\sqrt{n}c) = \lim_{n\to\infty} \Phi(t-\sqrt{n}c)$$
that I used above is correct or not. Any suggestion is welcomed.
