Bounded Below Subset of Real Numbers/Examples/Open Interval from 0 to Infinity

Example of Bounded Below Subset of Real Numbers

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 $0$.

Sources

• 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Real Numbers: $1.35$. Example $\text{(b)}$
• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 2$: Continuum Property: $\S 2.4$ Examples: $\text {(iii)}$