Proving the Existence of a Linear Functional on $\ell^{\infty}$ Bounded by $\limsup$

Question

Let $X=\ell^{\infty}$ and $p: X \rightarrow \mathbb{R}, p(x)=\limsup _{n \rightarrow \infty} x_n$. Is there a linear functional $f: X \rightarrow \mathbb{R}$ such that $$ -p(-x) \leq f(x) \leq p(x) ? $$

My Attempted Solution:

1. Sublinearity of $p$:

   • We verify that $p$ is a sublinear function. A function $p: X \rightarrow \mathbb{R}$ is sublinear if it satisfies:
     • (i) $p(x + y) \leq p(x) + p(y)$ for all $x, y \in X$,
     • (ii) $p(tx) = tp(x)$ for all $x \in X$ and $t \geq 0$.
   • For $p(x) = \limsup_{n \rightarrow \infty} x_n$, these properties hold because:
     • $\limsup_{n \rightarrow \infty} (x_n + y_n) \leq \limsup_{n \rightarrow \infty} x_n + \limsup_{n \rightarrow \infty} y_n$ for any $x, y \in \ell^{\infty}$,
     • $\limsup_{n \rightarrow \infty} (tx_n) = t \cdot \limsup_{n \rightarrow \infty} x_n$ for any $x \in \ell^{\infty}$ and $t \geq 0$.

2. Constructing a Linear Functional on a Subspace:

   • To apply the Hahn-Banach Theorem, we need a linear functional defined on a subspace of $\ell^{\infty}$ that is dominated by $p$.
   • Consider the subspace $Y$ of $\ell^{\infty}$ consisting of all sequences that eventually become constant. Define $f: Y \rightarrow \mathbb{R}$ by $f(y) = \lim_{n \rightarrow \infty} y_n$. For sequences in $Y$, the limit and the limit superior are the same.
   • $f$ is linear on $Y$, and for any $y \in Y$, $f(y) = \lim_{n \rightarrow \infty} y_n \leq \limsup_{n \rightarrow \infty} y_n = p(y)$.

3. Applying the Hahn-Banach Theorem:

   • By the Hahn-Banach Theorem, there exists an extension $F: X \rightarrow \mathbb{R}$ of $f$ to all of $\ell^{\infty}$ such that $F(x) \leq p(x)$ for all $x \in \ell^{\infty}$.
   • Moreover, since $F$ is linear, the inequality $-p(-x) \leq F(x) \leq p(x)$ holds for all $x in \ell^{\infty}$. This follows because for any $x \in X$, $-p(-x) = -\limsup_{n \rightarrow \infty} (-x_n) = \liminf_{n \rightarrow \infty} x_n \leq F(x)$, by the definition of $F$ and the properties of limit inferior and limit superior.

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EDIT: I edited the answer according to the comment, is it fine now?