This is a very simple question.
Suppose $E$ is a bounded set in $\mathbb{R}^n$. Let $\lambda>1$, and define $$E_{\lambda}:=\lambda E=\{\lambda x: x\in E \}.$$ We call $E \mapsto \lambda E$ the homothetic transformation by scaling factor $\lambda$.
Clearly from the standard picture, for example https://en.wikipedia.org/wiki/Homothetic_transformation, we can observe after a homothetic transformation by $\lambda>1$, the new set is just an enlargement of the original one. Although the new set doesn't necessarily contain the original set, after a translation we can really expect so. More precisely, the following claim should be true
Claim: Fix a bounded set $E \subset \mathbb{R}^n$, for any $\lambda>1$, there must exist a vector $ \overrightarrow{b} \in \mathbb{R}^n$ such that $E \subset \subset \lambda E+\overrightarrow{b}$.
I've no idea how to "mathematically" prove this claim. I strongly believe this is true. Can anyone write a rigorous proof? Thanks in advance!
As @coffeemath pointed out, the claim fails to be true if $E$ is a disconnected set. Let's suppose if $E$ is connected, is this claim true?
What if $E$ is two points at distance $1$ and $\lambda$ is quite large, so that $\lambda(E)$ is two other points which are very far apart? It would seem $E$ couldn't be contained in any translate of $\lambda E$ since such translates consist of two points at large distance from each other.