I am reading [Agler and McCarthy's Pick Interpolation and Hilbert Function spaces](https://www.math.wustl.edu/wp/mccarthy/pick-interpolation-and-hilbert-function-spaces/). In the Chapter 9, titled "Interpolating Sequences", the authors say that every interpolating sequence is strongly separated. Here's what it means to say a sequence is *interpolating sequence for $H^{\infty}({\mathbb D})$* and for it to be *strongly separated*:

**Definition (interpolating sequence).** A sequence of points $\{ \lambda _i \}_{i=1}^{\infty}$ in the open unit disc $\mathbb D$ is called *interpolating sequence for $H^{\infty}({\mathbb D})$* if for any bounded sequence $\{ w_i \}_{i=1}^{\infty}$ there is a function $\phi \in H^{\infty}({\mathbb D})$ such that $\phi (\lambda_i) = w_i$ for each $i$.

**Definition (strong separated sequence).** A sequence of points $\{ \lambda _i \}_{i=1}^{\infty}$ in the open unit disc $\mathbb D$ is called *strongly separated sequence for $H^{\infty}({\mathbb D})$* if there is some constant $\epsilon > 0$ such that for each $i$ there is a function $\phi_i \in H_1^{\infty}({\mathbb D})$, the closed unit ball of $H^\infty (\mathbb D),$ such that $\phi_i (\lambda_i) = \epsilon$ and $\phi_i (\lambda_j) = 0$ for all $j \ne i$.

The authors say that it can be shown that every interpolating sequence is strongly separated by closed graph theorem but I do not exactly see how. Here's my attempt, nonetheless:

Let $\{ \lambda _i \}_{i=1}^{\infty}$ in the open unit disc $\mathbb D$ be an interpolating sequence for $H^{\infty}({\mathbb D})$. Define a map $T : H^\infty (\mathbb D ) \to \ell ^\infty$ by $T(\phi ) = \{ \varphi (\lambda _i) \}_{i=1}^{\infty}$. This map is surjective by definition of interpolating sequences and I managed to show that $T$ is bounded by the closed graph theorem. 

Now we can take $\epsilon = \lVert T \rVert$ and consider the sequence $\{ \epsilon e_i \}_{i=1}^{\infty}$ in $\ell ^\infty$ . Since $T$ is surjective, there is some $\varphi _i \in H^{\infty} (\mathbb D)$ such that $T(\varphi_i) = \{ \epsilon e_i \}_{i=1}^{\infty}$. Now $\lVert T(\varphi _i) \rVert = \varepsilon \lVert e_i \rVert =\varepsilon$. But from here, I can only conclude that $\lVert \varphi_i \rVert \ge 1$ whereas I am looking for the reverse inequality.