Let $(X^N/N, N \in \mathbb{N}$) satisfy a large deviations principle in $\mathbb{R}$ with convex rate function $I$. Let $\alpha$ be a positive real number. Show that $X^{\lfloor \alpha N \rfloor}/N, N \in \mathbb{N}$ satisfies a large deviations principle in $\mathbb{R}$ with rate function $J(x) = \alpha I(x/\alpha)$.
The hint in the book ("Big Queues") says I am supposed to prove the upper bound for closed half spaces, then extend to general closed sets, prove the lower bound for small open balls, then extend to general open sets.
I tried to write something about the upper bound for $[x, \infty)$. Given that $I$ is a rate function for $X^N/N$, I know that $$\limsup_{N \to \infty} \frac{1}{N} \log P \left( \frac{X^N}{N} \geq x \right) \leq -\inf_{[x, \infty)} I(x)$$ how do I introduce $\alpha$ into this inequality?
I'm sorry if this question is stupid - I am very new to the fields of (rigorous) probability and large deviations, and the learning curve seems to be quite steep.