Midterm

[ceciliachan1979]

No description provided.

[E7-87-83]

Problem 2 嘅 presentation 可以做好啲
例如，我未見過 math book 咁寫

equal sign 可以「 indent」, 但greater than 、 less than 通常每行都寫曬兩邊

收返。有數學書

[E7-87-83]

去證一個負數 x 嘅absolute value 少於 A，通常咁做

Prove x < 0
Prove -x < A
Conclude |x| <A

去證一個正數 x 嘅absolute value 少於 A，通常咁做

Prove x > 0
Prove x < A
Conclude |x| <A

負數嗰部分，我地唔會喺line 2畫蛇添足寫成 -x < 0 < A

[E7-87-83]

Problem 5 - correct

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Prob 1

Prob 1.3: 問嘅係 p => q 係 true 定 false 。簡單答 false 即可。

Other parts are correct.

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Prob 3

Presentation 需要改善。
line 1: A < B < C
line 2: D < B < E

咁代表乜？ "A=D and C=E" 、" D<A and C<E" 、"A=D and C<E" 、"D<A and C=E" (仲有greater-than-or-equal-to-sign嘅cases，略)定係 line 1 跟 line 2 無因果關係？？

[E7-87-83]

Prob 7
idea 無錯，但presentation 可以更rigorous。例如第一段
"The Bolzano-Weierstrass theorem states that every bounded sequence has a
convergent subsequence. The sequence {a_n } is bounded, so it has a convergent
subsequence {a_{n_k} }."
我會改成
"The Bolzano-Weierstrass theorem states that every bounded sequence has a
convergent subsequence; in other words, if a sequence {a_n } is bounded, then it has a convergent
subsequence {a_{n_k} }."

suffix 係computer science term; 用返數學term, subsequence 。

[E7-87-83]

https://en.wikipedia.org/wiki/Language_of_mathematics

[ceciliachan1979]

Prob 3

Presentation 需要改善。 line 1: A < B < C line 2: D < B < E

咁代表乜？ "A=D and C=E" 、" D<A and C<E" 、"A=D and C<E" 、"D<A and C=E" (仲有greater-than-or-equal-to-sign嘅cases，略)定係 line 1 跟 line 2 無因果關係？？

line 1 去 line 2 只係全部 + 2

[E7-87-83]

Prob 4 OK, 但似乎頭昏用錯字，已經證明咗limit存在，但之後"Suppose the limit exists,"

另外，我會寫多一啲details

Since the limit exists, let the limit be $x_0$, then

\begin{eqnarray*}
x_0
& = & \lim_{n \to \infty} a_n \
&=& \lim_{n \to \infty} \sqrt{4 + a_{n-1}} \
&=& \sqrt{4 + \lim_{n \to \infty} a_{n-1}} \
&=& \sqrt{4 + x_0 } \
\end{eqnarray*}

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Prob 6 - OK

[ceciliachan1979]

去證一個負數 x 嘅absolute value 少於 A，通常咁做

Prove x < 0 Prove -x < A Conclude |x| <A

去證一個正數 x 嘅absolute value 少於 A，通常咁做

Prove x > 0 Prove x < A Conclude |x| <A

負數嗰部分，我地唔會喺line 2畫蛇添足寫成 -x < 0 < A

其實呢句係講緊邊度？

GitHub 個 review 可以俾你指住個 file 嘅 line number 嚟俾 review comment 㗎
你去個 Files Changed tab 嗰度，就可以睇到啲 tex source，highlight 一行 source，左手邊就會出現一個藍色加號，就可以係嗰度 comment。

[ceciliachan1979]

Prob 1

Prob 1.3: 問嘅係 p => q 係 true 定 false 。簡單答 false 即可。

Other parts are correct.

其實 problem 1.3 咁寫，主要係驚個 course instructor 淨係講 R，從來冇教過其它 metric space 同 metric space 唔一定 complete，in that case 答 false 就會錯，所以戴個頭盔啫 ⋯

[ceciliachan1979]

Prob 7 idea 無錯，但presentation 可以更rigorous。例如第一段 "The Bolzano-Weierstrass theorem states that every bounded sequence has a convergent subsequence. The sequence {a_n } is bounded, so it has a convergent subsequence {a_{n_k} }." 我會改成 "The Bolzano-Weierstrass theorem states that every bounded sequence has a convergent subsequence; in other words, if a sequence {a_n } is bounded, then it has a convergent subsequence {a_{n_k} }."

suffix 係computer science term; 用返數學term, subsequence 。

suffix 同 subsequence 係唔同㗎⋯
suffix 係 {a_7,a_8,a_9,...}，後面啲 elements 我全部都要
subsequence 可以係 {a_7, a_{10}, a_{23}, ...}，後面啲 elements 可以選擇性唔要

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suffix 同 subsequence 係唔同㗎⋯ suffix 係 {a_7,a_8,a_9,...}，後面啲 elements 我全部都要 subsequence 可以係 {a_7, a_{10}, a_{23}, ...}，後面啲 elements 可以選擇性唔要

OK. Apology.

[ceciliachan1979]

suffix 同 subsequence 係唔同㗎⋯ suffix 係 {a_7,a_8,a_9,...}，後面啲 elements 我全部都要 subsequence 可以係 {a_7, a_{10}, a_{23}, ...}，後面啲 elements 可以選擇性唔要

OK. Apology.

咁其實又唔駛道歉嘅﹐做錯數啫，我條條都有啲錯㗎啦，要道歉我咁多題做錯咪有排煩 😅 ⋯
尤其係其實係你啱，呢個位係應該用 subsequence