Let $H$ be a Hilbert space, $M \subset H$ a convex subset, and $\left( x_n \right)$ a sequence in $M$ such that $\left\lVert x_n \right\rVert \to d$, where $d = \inf_{x \in M} \lVert x \rVert$. How to show that $\left( x_n \right)$ converges in $H$?
What are the most surprising facts about the plane $\mathbb{R}^2$ and the space $\mathbb{R}^3$ that one can derive using the above result?
I think you mean $\|x_n\| \to d$. If $d=0$, it is trivial, so suppose $d>0$. By parallelogram law, $$ \|x_n - x_m\|^2 = 2\|x_n\|^2 + 2\|x_m\|^2 -4 \|(x_n+x_m)/2\|^2 $$ Since $M$ is convex, $(x_n + x_m)/2 \in M$, so $$ \|(x_n+x_m)/2\| \geq d $$ Now use the fact that $\|x_n\| \to d$ to conclude that $(x_n)$ is Cauchy and hence convergent.
About the "surprising fact", not sure what you are looking for.