Open covering with bounded diameters [closed]

Question

Here is an interesting puzzle I came across.

I have no idea which tools could be applied to solve it, so the tags may be misleading.

For any $A \subseteq \mathbb{R^n}$ , its diameter is defined by $$\mathrm{diam}A = \sup_{x,y \in A} |x-y|$$

Suppose $\{U_{i}\}_{i = 1,2,\cdots}$ is an open covering of the unit $n$-cube and the diameter of each $U_{i}$ is less than one. Show that there exists a point $x$ in the cube which lies in at least $(n+1)$ different $U_i's$

Any hint is highly appreciated!!!

Answer 1

The answer is given by the Lebesgue's Covering Theorem (numbered as 1.8.20 in Engleking's book "Theory of dimensions: finite and infinite"): If $\mathcal F$ is a finite closed cover of the $n$-cube $I^n$ no member of which meets two opposite faces of $I^n$, then $ord(\mathcal F)\ge n$ (the inequality $ord(\mathcal F)\ge n$ means that there is a point in $I^n$ that belongs to more than $n$ elements of the cover $\mathcal F$).