Answer to Question #207582 in Real Analysis for Sam

Question #207582

Check wether the collection G, given by:

G’ = {]1/(n+2), 1/n[ : n∈N}

is an open cover of ]0,1[.



1
Expert's answer
2021-06-18T12:24:23-0400

(0,1) has no finite subcover because,if possible it has a subcover

i.e then "U^n_{i=1}I_n \\eqslantgtr (0,1)"

"I_n=(\\frac{1}{n+2}, \\frac{1}{n})"

But "\\frac{1}{m+1}< \\frac{1}{n}" and "0<\\frac{1}{m+1}<1"

But "\\frac{1}{m+1} \\notin U^n_{i=1}I_n" So, "[I_n ]_{n=1}^m" is not a subcover

That is to say for every m , we see that "\\frac{1}{m+1} \\notin U_{n=1}^m I_n"

So it has no subcover


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