MHB Show that the sequence has a decreasing subsequence

mathmari
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Hi ! :)

Let x_{n} a sequence of positive numbers.How could I show that it has a decreasing subsequence that converges to 0,knowing that inf{x_{n},n ε N} =0??
 
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mathmari said:
Hi ! :)

Let x_{n} a sequence of positive numbers.How could I show that it has a decreasing sub-sequence that converges to 0,knowing that inf{x_{n},n ε N} =0??

I really think this theorem will help you:

Theorem:
A bounded sequence of \mathbb{R} has a convergent sub sequence.

If a sequence X is bounded,all its sub-sequences will be bounded. Now since every sequence has a monotone sub-sequence (i.e either decreasing or increasing), X will also have a monotone sub-sequence.

Therefore By Monotone Convergence Theorem the sub-sequence being bounded and Monotone will converge.

Your sequence is decreasing, its obvious it will tend to its infimum.
 
mathmari said:
Hi ! :)

Let x_{n} a sequence of positive numbers.How could I show that it has a decreasing subsequence that converges to 0,knowing that inf{x_{n},n ε N} =0??

If $\displaystyle \text{inf} [x_{n}] = 0$ and for all n is $\varepsilon > 0$ then by definition for a $\displaystyle \varepsilon > 0$ it exists at least an n for which is $\displaystyle x_{n} < \varepsilon$ and that means that for all n it exists at least one m for which is $\displaystyle x_{m} < x_{n}$...

Kind regards

$\chi$ $\sigma$
 
Ok!Thank you for your help! :)
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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