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--- math104-f21:final-mistakes [2021/12/16 02:41]
pzhou
+++ math104-f21:final-mistakes [2026/02/21 14:41] (current)
@@ Line 13: / Line 13: @@
 === 3 ===
-One need to show that $x_n$ is bounded from below; $x_n$ is monotone decreasing. This two show $x_n$ converges to some $x$. Then one need to prove that $x=\sqrt{a}$.
+One need to show that $x_n$ is bounded from below; $x_n$ is monotone decreasing. These two conditions together show $x_n$ converges to some $x$. Then one need to prove that $x=\sqrt{a}$. Missing any of the three steps would make one lose points.
+There are other methods to prove this problem, such as creating an auxillary sequence $y_1 = x_1, y_{n+1} = (y_n + \sqrt{a})/2$, and show that $x_n \leq y_n$ and $y_n \to \sqrt{a}$.
 === 4 ===

Lecture Notes