Let $\mathcal H$ a Hilbert space over $\mathbb R$ and $A = \{x\in \mathcal H : \langle x, a \rangle \geq 1 \}$. I'm trying to prove that $A$ is closed.
Let $(x_n) \subset A$ be a Cauchy-sequence. Since $\mathcal H$ is a Hilbert space, the sequence converges to an $x \in \mathcal H$.
Suppose the sequence $(\langle x_n, a \rangle)\subset \mathbb R$ is also Cauchy. Since $[1,\infty)$ is closed, its limit, $\langle x, a \rangle$, is also in $[1,\infty)$. So $x \in A$, and $A$ is closed.
However, why is the sequence $(\langle x_n, a \rangle)$ Cauchy whenever $(x_n)$ is?