Why does a monotone increasing sequence bounded above converge to its supremum?

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

. Due to the sequence's monotonicity, all subsequent terms must be greater than or equal to this specific term, trapping them within the interval

(L - \varepsilon, L]

. This satisfies the formal epsilon-N definition of convergence.

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.

Reasoning, step by step

Identify the supremum $L$ of the sequence values, which exists by the Least Upper Bound Property.
For any given $\varepsilon > 0$ , consider the value $L - \varepsilon$ .
Since $L$ is the *least* upper bound, $L - \varepsilon$ cannot be an upper bound.
Therefore, there must exist some term $a_N$ in the sequence such that $a_N > L - \varepsilon$ .
Use the monotonicity condition: for all $n > N$ , $a_n \ge a_N$ .
Combine these inequalities to show that for all $n > N$ , $L - \varepsilon < a_n \le L$ .
Conclude that the terms are trapped in the epsilon neighborhood of $L$ , proving $\lim(a_n) = L$ .

Example

The video visually demonstrates this by plotting a rising dotted curve for $\{a_n\}$ , marking the red dashed line for the supremum $L$ , and the green dashed line for $L - \varepsilon$ . It shows that once the curve crosses the green line at index $N$ , it remains trapped between the green and red lines for all subsequent indices.

Common misconceptions

Believing that the sequence must reach the supremum $L$ exactly at some finite term.
Thinking that the proof relies on the sequence being strictly increasing rather than just monotone increasing.
Confusing the supremum with a maximum value that must be attained by the sequence.