I'm having trouble filling the steps in this guided proof of Riesz's representation theorem. (I already have a proof I can understand, but I'd like to understand this one too.)

Let $H$ be a Hilbert space, and $\varphi : H \to \mathbb{C}$ a bounded linear functional. If $\varphi = 0$ then we are done; otherwise, by scaling, we may assume without loss of generality that $\| \varphi \| = 1$. So, for each $n$, there is a unit vector $h_n$ in $H$ such that $| \varphi(h_n) | > 1 - \frac{1}{n}$. By multiplying each one by an appropriate complex number of unit modulus, we may assume $\varphi(h_n) \in \mathbb{R}$ and $\varphi(h_n) > 1 - \frac{1}{n}$.

Now I run into a problem — I can see that everything follows from the first step, but the first step is eluding me at the moment.

1. $h_n \longrightarrow h$ for some $h$ in $H$. Why? If $H$ is finite-dimensional then certainly there is a convergent subsequence, but I don't see how we can assert the existence of such an $h$ without knowing more about the relative distances of the $h_n$.

2. $h$ is orthogonal to the kernel of $\varphi$. I think the idea here is to show that $\| h - u \|$ is minimised over $u \in \ker \varphi$ when $u = 0$, by exploiting the fact that $\| h - u \| \ge | \varphi(h - u) | = | \varphi(h) | = 1$.

3. $\ker \varphi \oplus \operatorname{span} \{ h \} = H$, by e.g. rank–nullity or orthogonal decomposition.

4. Hence $\varphi(x) = \langle x, h \rangle$ for all $x \in H$, by decomposing $x$ using the above decomposition of $H$ and linearity.