Answers to homework set #2

Student-provided answers to homework set #2, due date unspecified (not to be handed in and thus not marked). The content on this page is solely intended to function as a study aid for students and should constitute fair dealing under Canadian copyright law.

Problems to complete: all problems in (4.1), (4.2), (4.3) (but 4.3 is not emphasized).

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1Section 4.1¶

1.1Question 1¶

Determine a condition on $|x-1|$ that will ensure that
(a) $|x^2-1| < 1/2$

$|x^2-1| = |x-1||x+1|$. If $|x-1| < 1$, then $|x| \leq 2$ so $|x+1| \leq |x| + 1 \leq 3$. Then $|x^2-1| \leq 3|x-c|$. Let $\epsilon = 1/2$. Then if $|x-c| < \epsilon/3$, $|x^2-1| \leq \epsilon = 1/2$.

(b) $|x^2-1| < 1/10^3$ (I think there's a typo in my version - it says $1/10^{-3}$ but that is strange notation)

TBC

(c) $|x^2-1| < 1/n$
(d) $|x^3-1| < 1/n$

2Section 4.2¶

2.1Question 1¶

Trivial, not worth it

2.2Question 2¶

Soon

3Section 4.3¶

3.1Question 1¶

Prove Theorem 4.3.2

No thanks