Proof of Cramer's Large Deviation Theorem

Question

https://ocw.mit.edu/courses/sloan-school-of-management/15-070j-advanced-stochastic-processes-fall-2013/lecture-notes/MIT15_070JF13_Lec4.pdf

On page 6 (proof of Cramer's theorem), it says

$$\limsup_n \frac{1}{n} \log x_n \leq - \inf_{x\in F} I(x)$$

$$\limsup_n \frac{1}{n} \log y_n \leq - \inf_{x\in F} I(x)$$

implies

$$\limsup_n \frac{1}{n} \log(x_n + y_n) \leq - \inf_{x\in F} I(x).$$

I don't see why this is true and I'm struggling to prove this. Why does that last inequality follow from the previous two?

Answer 1

There's a small trick when you want crude bounds: bound the sum with the maximum to put it outside. Here:

$$\frac{\ln (x_n+y_n)}{n} \leq \frac{\ln (2 \max \{x_n, y_n\} )}{n} = \frac{\max \{\ln (x_n), \ln(y_n)\}}{n}+\frac{\ln (2)}{n} \leq \max \left\{\frac{\ln (x_n)}{n}, \frac{\ln(y_n)}{n}\right\}+\frac{\ln (2)}{n}.$$

Now, take the limsup.