Monotone convergence from the supremum principle

The red dashed line represents the supremum $L$, which serves as the least upper bound for the sequence $\{a_n\}$. Visually, it establishes the ceiling that the sequence approaches but never exceeds.

Conditions: Visual demonstration of a monotone increasing sequence.; Existence of a supremum $L$.

A monotone increasing sequence bounded above converges to its supremum because the supremum acts as the least upper bound. For any arbitrary tolerance $\varepsilon > 0$, the property of the supremum guarantees that there exists at least one term in the sequence that exceeds $L - \varepsilon$.

Conditions: The sequence $\{a_n\}$ is monotone increasing.; The sequence $\{a_n\}$ is bounded above.; $L$ is the supremum of the set of sequence values.; $\varepsilon > 0$ is an arbitrary positive tolerance.

First, the supremum property ensures there exists an index $N$ such that $a_N > L - \epsilon$. Second, monotonicity ensures that for all $n > N$, $a_n \ge a_N$, so $a_n > L - \epsilon$.

Conditions: Sequence $\{a_n\}$ is monotone increasing.; Sequence $\{a_n\}$ is bounded above by $L$.; $\epsilon > 0$ is given.

By definition, a monotone increasing sequence satisfies $a_{n+1} \ge a_n$ for all $n$. Through induction or transitivity, if $n > N$, then $a_n \ge a_{n-1} \ge \dots \ge a_N$.

Conditions: The sequence $\{a_n\}$ is monotone increasing.; $N$ is a fixed integer index.; $n$ is any integer such that $n > N$.

The video uses an animated Cartesian coordinate system. The horizontal axis represents the index $n$, and the vertical axis represents the term values $a_n$.

Conditions: The visualization assumes a monotone increasing sequence bounded above.; The axes are labeled for index and value.

The Least Upper Bound Property guarantees the existence of a unique supremum $L$ for the set of sequence values, provided the set is non-empty and bounded above. This supremum $L$ serves as the candidate limit for the sequence.

Conditions: The set of sequence values is non-empty.; The set of sequence values is bounded above.

This double inequality is valid for all indices $n$ such that $n > N$, where $N$ is the specific index guaranteed by the supremum property such that $a_N > L - \epsilon$. Before $N$, terms may be less than or equal to $L - \epsilon$.

Conditions: $N$ is defined such that $a_N > L - \epsilon$.; $n$ is an integer index.; Sequence is monotone increasing and bounded above by $L$.

The green dashed line indicates the value $L - \epsilon$ for any given $\epsilon > 0$. It visually demonstrates the concept of an 'epsilon neighborhood' below the supremum.

Conditions: $\epsilon > 0$ is an arbitrary positive distance.; $L$ is the supremum.

The definition of supremum includes two conditions: $L$ is an upper bound, and no number smaller than $L$ is an upper bound. If we take any $\varepsilon > 0$, the value $L - \varepsilon$ is strictly less than $L$.

Conditions: $L$ is the supremum of the sequence values.; $\varepsilon > 0$ is an arbitrary positive number.

Monotonicity ensures that once a term $a_N$ exceeds $L - \varepsilon$, all subsequent terms $a_n$ (for $n > N$) are greater than or equal to $a_N$. Since the sequence is also bounded above by $L$, these terms satisfy $L - \varepsilon < a_n \le L$.

Conditions: The sequence is monotone increasing.; There exists an index $N$ such that $a_N > L - \varepsilon$.; $L$ is an upper bound for the sequence.

The Least Upper Bound Property defines $L$ as the *smallest* number that bounds the set from above. If we subtract any positive amount $\epsilon$ from $L$, the resulting value $L - \epsilon$ is strictly less than the least upper bound.

Conditions: $L$ is the supremum (least upper bound) of the set $\{a_n\}$.; $\epsilon > 0$ is an arbitrary positive real number.