Recent content by xento

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    Show that if a ∈ R, then: sup{r ∈ Q : r < a} = a

    What I find hard from studying real analysis is that the type of questions can vary tremendously, and there are only limited examples in textbooks. Also, it is not like doing calculus for example where you do problems, check the answers and build up your confidence as you get more and more right...
  2. X

    Show that if a ∈ R, then: sup{r ∈ Q : r < a} = a

    I'm pretty lost myself. Here's another idea I had. Density Theorem: if x and y are any real numbers with x < y , then there exists a rational number r \in \bold{Q} such that x < r < y. Let X=\{x \in \bold{R} : x < a \} Let Y=\{r \in \bold{Q} : r < a \} If a is an upper bound of X such...
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    Show that if a ∈ R, then: sup{r ∈ Q : r < a} = a

    Would it work if we proved: \sup \{x \in \bold{R} : x < a \}=a and then just state that since \{r \in \bold{Q} : r < a \} is a subset of \{x \in \bold{R} : x < a \} The proof is complete?
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