$\{x_n\}$ is nonnegative. Show $\limsup_{n \to \infty} {something} \le \limsup_{n \to \infty} {another thing}$. Also give examples of $\{x_n\}$.

Question

Let $\{x_n\}$ be bounded below by $0$. Prove

$$\limsup_{n \to \infty} {(x_{n + 2} + (n + 2)x_{n + 1})\over 2n + 3} \le \limsup_{n \to \infty} {(x_{n + 2} + (n + 1)(x_{n + 1} - x_n))\over 2}$$

I've tried 'proof by contradiction' but it fails.

On the other hand, say each $x_n = n$, then with this $\{x_n\}$, is the LHS of the equation $\lt$ the RHS? Thanks a lot.