MHB Sequences and their limits, convergence, supremum etc.

ertagon2
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Could someone check if my answers are right and help me with question 5?
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Are you sure about the third statement in 4?
Also, you probably have some idea about 5. Are you doubting between two possibilities?
 
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Thread '2-sphere intrinsic definition by gluing disks' boundaries'

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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