On dense embedding of Banach spaces

Great question! What you need is Sandy Grabiner's approximation lemma:

Lemma: Let $E$ and $F$ be Banach spaces, let $T \in B(E,F)$, and let $M > 0$ and $0 < r < 1$. Suppose that for each $y \in [F]_1$ there exists $x_0 \in [E]_M$ with $\|y - Tx_0\| \leq r$. Then for each $y \in F$ there exists $x \in E$ with $\|x\| \leq \frac{M\|y\|}{1-r}$ and $Tx = y$.

I think this immediately implies that $F = E$ in your question. You can probably prove the lemma yourself very easily (hint: geometric series), but for the sake of completeness a reference is: Theorem 3.35 of my book Measure Theory and Functional Analysis.

It's a great lemma and deserves to be better known. For instance, both the open mapping theorem and Tietze's extension theorem follow easily from it.