Bounded Below Subset of Real Numbers/Examples
Examples of Bounded Below Subsets of Real Numbers
Example: $\openint 1 \to$
The subset $S$ of the real numbers $\R$ defined as:
- $S = \openint 1 \to$
is bounded below.
Examples of lower bounds of $S$ are:
- $-7, 1, \dfrac 1 2$
The set of all lower bounds of $S$ is:
- $\hointl {-\infty} 1$
Example: $\set {x \in \R: x > 0}$
The subset $T$ of the real numbers $\R$ defined as:
- $T = \set {x \in \R: x > 0}$
is bounded below, but unbounded above.
Let $H > 0$ in $T$ be proposed as an upper bound.
Then it is seen that $H + 1 \in T$ and so $H$ is not an upper bound at all.
Examples of lower bounds of $T$ are:
- $-27, 0$
Its infimum is $0$.
Example: $\hointl {-\infty} 2$
Let $I$ be the open real interval defined as:
- $I := \openint 0 1$
Then $I$ is not bounded below.
Hence $I$ does not admit an infimum, and so does not have a smallest element.
Example: $\openint 0 1$
Let $I$ be the open real interval defined as:
- $I := \openint 0 1$
Then $I$ is bounded below by, for example, $0$, $-1$ and $-2$, of which $0$ is the infimum.
However, $I$ does not have a smallest element.
Example: $\set {-1, 0, 2, 5}$
Let $I$ be the set defined as:
- $I := \set {-1, 0, 2, 5}$
Then $I$ is bounded below by, for example, $-1$, $-2$ and $3$, of which the infimum is $-1$.
$5$ is also the smallest element of $I$.
Example: $\openint 3 \infty$
Let $I$ be the unbounded open real interval defined as:
- $I := \openint 3 \to$
Then $I$ is bounded below by, for example, $3$, $2$ and $1$, of which the infimum is $3$.
However, $I$ does not have a smallest element.
Example: $\closedint 0 1$
Let $I$ be the closed real interval defined as:
- $I := \closedint 0 1$
Then $I$ is bounded below by, for example, $0$, $-1$ and $-2$, of which $0$ is the infimum.
$I$ is also the smallest element of $I$.
Example: $\openint \gets \to$
Let $\R$ denote the set of real numbers.
$\R$ is not bounded below.
Example: $\openint \gets 0$
Let $I$ be the unbounded open real interval defined as:
- $I := \openint \gets 0$
Then $I$ is unbounded below.
Example: $\hointl 0 1$
Let $I$ be the left half-open real interval defined as:
- $I := \hointl 0 1$
Then $I$ is bounded below.
$0$ is an lower bound of $I$.