Limsup, liminf, and limit points (from Analysis I by T. Tao) (Part 2)

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In this part we complete the section 6.4; we will solve the exercises next Sunday. And tomorrow Gallian's algebra.
  1. A Mathematical Room
  2. Real analysis
  3. Limits of sequences
  4. Limit superior
  5. Limit inferior
  6. Limit point
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Friend @A Mathematical Room. Let me ask you a question please. At minute 35:50 of this video you write: "Next we have y < L^(-)=sup(a-_N)_N=m" and then you say "supremum means least upper bound". At the beginning of Proposition 6.4.12 it is mentioned that L^(+) and L^(-) are extended real numbers. So if the set {a-_N : N greater than or equal to m} has upper bound then, as you said sup(a-_N)=the least upper bound of {a-_N : N greater than or equal to m} and the result follows. But what if {a-_N : N greater than or equal to m} does not have upper bound? In this case sup(a-_N)=+∞ and then we cannot say that y is less than the least upper bound. So in case L^(-), L^(+)∈{+∞, -∞} how can we get the result?

vichg