Prove that lim sup(x_n) = max(lim sup(y_n), lim sup(z_n))

[drawar]

Homework Statement

Let (x_{n}) be a bounded sequence. For each n \in \mathbb{N}, let y_{n}=x_{2n} and z_{n}=x_{2n-1}. Prove that
\lim \sup {x_n} = \max (\lim \sup {y_n},\lim \sup {z_n})

Homework Equations

The Attempt at a Solution

Don't know if I'm at the right path but I've tried letting M= \lim \sup {x_n}, M_{1}= \lim \sup {y_n}, and M_{2} = \lim \sup {z_n} and see that M \geq \max (M_{1}, M_{2}). How do I proceed from here to prove that M = \max (M_{1}, M_{2})? Thank you!

 

[funcalys]

Does this observation help?
$\sup {x_n} = \max (\sup {y_n},\sup {z_n})$