Q1 Let $f: [0, \infty)\rightarrow \mathbb{R}$ be a continuous function. Prove that there exist a sequence of real numbers satisfying
(a) $M_n \leq M_{n+1}$ for all $n\in \mathbb{N}$
and (b) $f(x)\leq M_n$ for all $x\in [0, n]$
Q2 Given a continuous function $f: [0, \infty)\rightarrow \mathbb{R}$, Is it always possible to construct $\{M_n\}$ which satisfies both the condition (a) and (b) in Q1 and (c) $\{M_n\}$ is a bounded sequence?
For Q1, I consider $M_n=\sup f(x)$ for $x\in [0, n]$ then the above two condition hold.
For Q2, If $\{M_n\}$ is a bounded sequence, then $\{M_n\}$ is convergent (Using (a) of Q1). But if we take $f(x)=x$ then $\{M_n\}=n$ which is divergent
Are the answers are correct? If something is wrong please correct me.