Let $X$ be a metric space and let $A\subset X$ be a bounded subset of $X$. I read on Wikipedia that the Hausdorff- and Kuratowski measures of non-compactness ($\alpha$, resp. $\beta$) satisfy the inequalities $$\alpha(A)\leq \beta(A)\leq 2\cdot\alpha(A)$$ How can I prove these inequalities?
How to show $\alpha(A)\leq \beta(A)\leq 2\alpha(A)$
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functional-analysis
metric-spaces
banach-spaces