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 be the distance from x to C. Since C is closed, d>0. For and , , so that . That shows that the map is continuous from C+Fx to F.
Now suppose that is a convergent sequence in C+Fx, with . (We want to show that .) Then as . Deduce from the continuity of that is Cauchy in F, hence converges to say (presumably the scalar field F is meant to be complete). It then follows that say, as . But C is closed, so it follows that . Finally, , so as required.