sets that do not have an infimum
Some examples for sets that do not have an infimum^{}:
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A more interesting example: The set ${M}_{2}:=\{x\in \mathbb{Q}:{x}^{2}\ge 2,x>0\}$ (again as a subset of $\mathbb{Q}$) .
Proof.
Clearly, $inf({M}_{2})>0$. Assume $i>0$ is an infimum of ${M}_{2}$. Now we use the fact that $\sqrt{2}$ is not rational, and therefore $$ or $i>\sqrt{2}$.
If $$, choose any $j\in \mathbb{Q}$ from the interval $(i,\sqrt{2})\subset \mathbb{R}$ (this is a real interval, but as the rational numbers^{} are dense (http://planetmath.org/Dense) in the real numbers, every nonempty interval in $\mathbb{R}$ contains a rational number, hence such a $j$ exists).
Then $j>i$, but $$, hence $$ and therefore $j$ is a lower bound for ${M}_{2}$, which is a contradiction^{}.
On the other hand, if $i>\sqrt{2}$, the argument^{} is very similar: Choose any $j\in \mathbb{Q}$ from the interval $(\sqrt{2},i)\subset \mathbb{R}$. Then $$, but $j>\sqrt{2}$, hence ${j}^{2}>2$ and therefore $j\in {M}_{2}$. Thus ${M}_{2}$ contains an element^{} $j$ smaller than $i$, which is a contradiction to the assumption^{} that $i=inf({M}_{2})$ ∎
Intuitively speaking, this example exploits the fact that $\mathbb{Q}$ does not have “enough elements”. More formally, $\mathbb{Q}$ as a metric space is not complete^{} (http://planetmath.org/Complete). The ${M}_{2}$ defined above is the real interval ${M}_{2}^{\prime}:=(\sqrt{2},\mathrm{\infty})\subset \mathbb{R}$ intersected with $\mathbb{Q}$. ${M}_{2}^{\prime}$ as a subset of $\mathbb{R}$ does have an infimum (namely $\sqrt{2}$), but as that is not an element of $\mathbb{Q}$, ${M}_{2}$ does not have an infimum as a subset of $\mathbb{Q}$.
This example also makes it clear that it is important to clearly state the superset^{} one is working in when using the notion of infimum or supremum.
It also illustrates that the infimum is a natural generalization^{} of the minimum of a set, as a set that does not have a minimum may still have an infimum (such as ${M}_{2}^{\prime}$).
Of course all the ideas expressed here equally apply to the supremum, as the two notions are completely analogous (just reverse all inequalities^{}).
Title  sets that do not have an infimum 

Canonical name  SetsThatDoNotHaveAnInfimum 
Date of creation  20130322 13:09:51 
Last modified on  20130322 13:09:51 
Owner  sleske (997) 
Last modified by  sleske (997) 
Numerical id  9 
Author  sleske (997) 
Entry type  Example 
Classification  msc 06A06 
Related topic  InfimumAndSupremumForRealNumbers 