Let X be a normed vector space. If C is a closed subspace x is a point in X not in C, show that the set C+Fx is closed. (F is the underlying field of the vector space).
Let $\displaystyle d = \inf\{\|x-c\|:c\in C\}$ be the distance from x to C. Since C is closed, d>0. For $\displaystyle c\in C$ and $\displaystyle 0\ne\lambda\in F$, $\displaystyle \|c+\lambda x\| = |\lambda|\|\lambda^{-1}c +x\|\geqslant d|\lambda|$, so that $\displaystyle |\lambda|\leqslant d^{-1}\|c+\lambda x\|$. That shows that the map $\displaystyle \phi:c+\lambda x\mapsto\lambda$ is continuous from C+Fx to F.
Now suppose that $\displaystyle (c_n+\lambda_n x)$ is a convergent sequence in C+Fx, with $\displaystyle \textstyle\lim_{n\to\infty}c_n+\lambda_n x = y\in X$. (We want to show that $\displaystyle y\in C+Fx$.) Then $\displaystyle c_m-c_n + (\lambda_m-\lambda_n)x\to0$ as $\displaystyle m,n\to\infty$. Deduce from the continuity of $\displaystyle \phi$ that $\displaystyle (\lambda_n)$ is Cauchy in F, hence converges to $\displaystyle \lambda_0$ say (presumably the scalar field F is meant to be complete). It then follows that $\displaystyle c_n\to y-\lambda x = c_0$ say, as $\displaystyle n\to\infty$. But C is closed, so it follows that $\displaystyle c_0\in C$. Finally, $\displaystyle c_n+\lambda_n x \to c_0+\lambda_0x$, so $\displaystyle y = c_0+\lambda_0x \in C+Fx$ as required.