Proof of the monotone convergence theorem

This video visually demonstrates the proof that every bounded monotonic sequence converges, utilizing the Least Upper Bound Principle. It begins by plotting a monotonically increasing sequence approaching a horizontal asymptote labeled as the supremum $L$. By introducing an arbitrary positive number $\varepsilon$, it establishes that $L-\varepsilon$ cannot be an upper bound, ensuring at least one term exceeds this value. Leveraging the sequence's monotonicity, all subsequent terms remain within the interval $(L-\varepsilon, L]$, thereby satisfying the formal definition of convergence to limit $L$.

Before you watch

• Real Analysis Fundamentals
• Definition of Sequence Limits