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Math and science::Analysis::Tao::06. Limits of sequences
Monotone bounded sequences converge
Increasing, decreasing and monotone
A sequence \( (a_n)_{n=0}^{\infty} \) is increasing
if \( a_n \le a_{n+1} \) for all \( n \in \mathbb{N} \) and decreasing
if \( a_n \gt a_{n+1} \) for all \( n \in \mathbb{N} \). A sequence is
monotone if it is either increasing or decreasing.
Monotone bounded sequences converge
If a sequence is monotone and bounded, then it converges.
The easiest card ever. The proof is on the reverse; can you think of it? The proof is easy and obvious in retrospect, but it uses a property that I originally didn't consider using to solve the problem.
Source
Abbott, p57
Tao, p138
