Let sequence of subsets of Real numbers $A_n$, $n\geq 1$ converge to $A$. Does the sequence of Inf $A_n$ converge to Inf$A$? The sequence need not be monotonic.
Sequence of Subsets of Real numbers converge implies their infimums also converge
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sequences-and-series
elementary-set-theory
convergence
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0If the sequence is not monotonic, then what does it mean for it to "converge" to $A$? – 2017-02-21
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1More generally, what does it mean for a sequence of subsets of reals to converge. – 2017-02-21
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0A sequence of sets $A_n$ converge to a set $A$ means that its indicator function converge pointwise. You can check this that for monotonic sets, it happens to be instersection/unions. – 2017-02-21
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No -- consider for example $$ A_n = (0,\tfrac1n) \cup \{1\} $$ This converges to $\{1\}$, but each $A_n$ has infimum $0$.
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0I think it converges to $\{1\}$ in the sense of pointwise convergence of the characteristic functions, but to $\{0,1\}$ with respect to the Hausdorff distance? Might be nice to get some clarification from the OP as to what sort of "convergence" he means. – 2017-02-21