I'm reading about thightness of probability measure sequences. The definition of the book is the next:
Def. A sequence $\{\mu_{n}\}$ of probability measures on $(\mathbb{R}^{k},\mathcal{B({\mathbb{R}^{k}}})$ is tight if for every $\epsilon>0,$ there is a bounded rectangle $A$ such that $\mu_{n}(A)>1-\epsilon$ for all $n.$ Here a bounded rectangle has the form $A=\{x\in\mathbb{R}^{k}: a_{i}<x_{i}\leq b_{i}, i=1,\ldots,k\}.$
I've found in other books (mostly ) the next definition:
A sequence $\{\mu_{n}\}$ of probability measures on $(\mathbb{R}^{k},\mathcal{B(\mathbb{R}^{k}})$ is thigt if for every $\epsilon>0,$ there is a compact set $K$ such that $\mu_{n}(K)>1-\epsilon$ for all $n.$
Are they equivalent definitions?
I'm trying to prove both definitions are equivalent but I'm stuck. My guess is covering $K$ with a finite collection of open rectangles which are contained in bounded rectangles and using the first definition, but I don't get any clear.
For the other direction I suppose that It is possible to approach any bounded rectangle with compact sets and to use the second definition (same idea as above) but I'm not sure.
Any kind o help is thanked in advanced.