The supremum of the set $X = \{x \in\mathbb{R} : x^2 < 7\}$ is $\sqrt7$.

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It's the exact same as with $\sqrt {2}$.

for positive $a,b$ we know $a < b \iff a^2 < b2$ so $X$ is a "cut", i.e. ever element of $X$ is less than any element not in $X$.

So $X$ is not empty and bounded above by ... say $3$ (or anything whose square is greater than $7$). So $x = \sup X$ exists. If $x^2 < 7$ then $x \in X$ and $x = \max X$. If we can prove $X$ has no maximum element we know that is not possible and $x^2 \ge 7$. If we can prove that for any $y|y^2 > 7$ then there is a $w < y| w^2 > 7$ then that means if $x^2 > 7$ then $x$ is not a least upper bound. The $x^2 = 7$.

Okay, So if $s^2 < 7$ we need to fine $t > s$ and $t^2 < 7$.

The usual trick is to find a $d| 0 < d < 7 - s^2$ and then $(s+ d)^2 = s^2 + 2ds + d^2 < 7$ so $2ds + d^2 < 7 - s^2$. We we can get $d = \frac {d'}s$ and $s > 1$ and $d' \le 1$ then we'd have $2ds + d^2 = 2d' + \frac {d'^2}{s^2} \le 2d' + d'^2 < 2d' + d' = 3d'$. So... heck with it... $2^2 = 4 < 7$ so we can assume $s \ge 2$ and and so $7 - s^2 \le 7-4 = 3$ so if $0< d' = \frac {7-s^2} 3< 1$ and $d = \frac {7-s^2}{3s}$ we are done.

$(s + d)^2 = s^2 + 2sd + d^2 = s^2 + 2s\frac {d'}s + \frac {d'^2}{s^2}$

$< s^2 + 2d' + d'^2 \le s^2 + 2d' + d' = s^2 + 3d'$

$= s^2 + 3\frac {7-s^2}3 = s^2 + 7 - s^2 = 7$.

So $X$ has no maximum. And $x= \sup X \not \in X$ so $x^2 \ge 7$.

We can do the same for $t^2 > 7$

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Updated on May 19, 2020

Comments

  • amWhy
    amWhy over 3 years

    I'm able to show that $\sqrt2$ is the supremum of $X = \{x \in\mathbb{R} : x^2 < 2\},\;$

    but I'm having trouble with this set with the set in the title.

    • Display name
      Display name over 6 years
      Just prove that the supremum of $X = {\{x ∈ R : x^2 < a}\}$ is $√a$ and then set $a=7$.
    • Reese Johnston
      Reese Johnston over 6 years
      Can you elaborate on how you showed it for $2$ rather than $7$? For most proof strategies, the $2$ shouldn't have been particularly relevant, so it should work exactly the same for $7$.
    • copper.hat
      copper.hat over 6 years
      The title is false.
    • egreg
      egreg over 6 years
      If you are able to prove that the supremum is $\sqrt{2}$, how can it be $\sqrt{7}$? A set has at most one supremum.
    • amWhy
      amWhy over 6 years
      The asker's question/title, was exactly what we see in the title now; an earlier edit (by @Rafael ) changed the title to ask $\sup x^2 \lt 2$ is $\sqrt 7$.
    • Display name
      Display name over 6 years
      Looks like OP changed the question to fix any errors. Anyways, my earlier comment still stands.
    • fleablood
      fleablood over 6 years
      If you actually prove that for all $b > 0$ then there exist a unique $a>0$ such that $a^2 = b$ and we call this $a = \sqrt b$ then it is trivial to show that $\sqrt 7$ is an upper bound of $X$ and that if $y < \sqrt{7}$ is not an upper bound (as $y < w < \sqrt{7}$ would imply $w \in X$) so $\sqrt{7} = \sup X$. We usually use the excercise $\sqrt{2}$ exists as means of introducing concepts. Then you prove $\sqrt {b}$ always exist. Then the question is moot.
  • Vivek Kaushik
    Vivek Kaushik over 6 years
    What's your proof that $\sup(-\sqrt{a},\sqrt{a}) =\sqrt{a}$ ?
  • Mikkel Rev
    Mikkel Rev over 6 years
    Suppose that $a >0$, then $\sqrt a$ is an upper bound. To see this, use the defintion of an open ball. Also $0 \in (-\sqrt a, \sqrt a)$ since $|0| < \sqrt a$. Therefore supremum exists. Now assume for contradiction that $\sqrt a$ is not the least upper bound. Then there exist $M \in \mathbb R$ which is the supremum and $M < \sqrt a$. Consider $z:=\frac{\sqrt a - M}{\sqrt a} + M$. By construction $z>M$. It is impossible that $z<\sqrt a$ since $M$ is the suprermum, But if $z \geq \sqrt a$, then $\sqrt a \leq \frac{\sqrt a - M}{\sqrt a} + M \implies \sqrt a \leq M$, a contradiction