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Atomised
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Homework Statement
Let [itex]\Lambda[/itex] = N and set A[itex]_{j}[/itex] = [j, [itex]\infty[/itex]) for j[itex]\in[/itex] N Then
j=1 to [itex]\infty[/itex] [itex]\bigcap[/itex] A[itex]_{j}[/itex] = empty set
Explanation: x[itex]\in[/itex] j=1 to [itex]\infty[/itex] [itex]\bigcap[/itex] provided that x belongs to every A[itex]_{j}[/itex].
This means that x satisfies j <= x <= j+1, [itex]\forall[/itex] j[itex]\in[/itex]N. But clearly this fails whenever j is a natural number strictly greater than x. In other words there are no real numbers which satisfy this criterion.
Homework Equations
I understand the importance of demonstrating that x belongs to Aj for all j
The Attempt at a Solution
Why not just choose x = j+1, thus it will belong to Aj
Homework Statement
I know this contradicts the x <= j+1 condition but I do not understand this condition, why can't x exceed j+1?
Apologies if formatting unclear.
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