MATH 3150 Class Notes 02-25-2025

notereal-analysis

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We proved the Bolzano-Weierstrass Theorem.

If $(x_n{)}_n$ is bounded, then by Bolzano-Weierstrass, we have that $x_{n_k}\to L$ for some subsequence $x_{n_k}$.

However, if $(x_n{)}_n$ is bounded and monotone, then by the Monotone Convergence Theorem, we have that $x_n\to L$.

Example

$$\{\begin{array}{l}{x}_{n+1}=1+\frac{1}{x_n},\\ x_1=1.\end{array}$$

Guess the limit: $L=1+\frac{1}{L}$ so $L^2-L-1=0$, and $L=\frac{1\pm \sqrt{5}}{2}$, and thus $L=\frac{1+\sqrt{5}}{2}$ since $x_n>0$.