### In-class translation exercise

## 國家太空中心改制行政法人　肩負太空發展任務

「國家太空中心設置條例」於1月1日施行，原隸屬於財團法人國家實驗研究院之國家太空中心，正式改制為行政法人國家太空中心，成為國家科學及技術委員會(國科會)轄下新設的法人機構。

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[Umberto]
The Act fot the Establishment of the Space Agency came into effect on January 1, 2023. The National Space Organization, previously an administrative branch of the National Applied Research Laboratory, was officially reorganized and renamed the Taiwan Space Agency, becoming a newly established legal entity under the National Science and Technology Council.

[Jason]The Act for National Space Agency Establishment came into effect on January 1. Taiwan Space Agency (TASA) which was subordinated to the incorporated legal entity National Applied Research Laboratory (NARLabs), is now stapled to the incorporated administrative legal entity : Taiwan Space Agency. It becomes the newly-established administrative legal entity under the control of National Science and Technology Council (STC).
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[Arren] The Act for the Establishment of the Taiwan Space Agency came into effect on January 1, and the National Space Organization, which was originally under the National Applied Research Laboratories, was officially transformed into Taiwan Space Agency, becoming a new legal entity under the National Science and Technology Council (NSTC).
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[Kathy] The Act for the Establishment of the Taiwan Space Agency officially went into effect on Jan. 1, 2023. The Taiwan Space Agency (TASA), formerly named the National Space Organization (NSPO), which was part of National Applied Research Laboratories (NARLabs), is now a newly established legal entity under the National Science and Technology Council (NSTC).
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[Sarah] Act for the Establishment of the Taiwan Space Agency has come into force since January 1. Taiwan's national space agency, which was placed under National Applied Research Laboratories, as officially upgraded with its new name, Taiwan Space Agency (TASA). It became a new entity under National Science and Technology Council (NSTC).
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[Sunny]The Act for the Establishment of the Taiwan Space Agency officially came into force on January 1.The Taiwan Space Agency, formerly known as the National Space Organization under the National Applied Research Laboratories, is now a new legal entity directed by Taiwan's National Science and Technology Council.
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[Daisy] The Act for the Establishment of the Taiwan Space Agency was enforced on January 1. Taiwan Space Agency (TASA), which was previously affiliated with National Applied Research Laboratories, is now officially restructured as a non-departmental public body and became a newly established legal entity under the National Science and Technology Council (NSTC).
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[Sofia] The Act for the Establishment of the Taiwan Space Agency takes effect from January 1. The National Space Organization, which was formerly affiliated with the National Applied Research Laboratories, has officially reformed the Taiwan Space Agency and is now an established legal entity under the jurisdiction of the National Science and Technology Council.
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[Wayne] The Act for the Establishment of the Taiwan Space Agency was implemented on January 1, 2023. National Space Agency, an institute affiliated with National Applied Research Labortays (NAR Labs), was rebranded as Taiwan Space Agency (TASA) and restructured as a legal institute under National Science and Technology Council (NSTC).
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[Teika] The Act for the Establishment of the Taiwan Space Agency took effect from January 1, 2023. Originally belonging to National Applied Research Laboratories (NARLabs), the formerly National Space Organization is now transformed into Taiwan Space Agency (TASA) and becomes a legal entity under Taiwan’s National Science and Technology Council (NSTC).
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國科會主任委員表示，行政院全力支持國家太空科技發展，國科會在政策面、計畫面、法制面與產業面攜手各部會與產業界布局我國太空發展藍圖，行政法人國家太空中心的成立，展現我國積極邁入太空發展的新紀元，以及臺灣在全球太空場域競逐的企圖心。

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[Umberto]
The minister of N remarked that

[Jason]
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[Kathy] Tsung-Tsong Wu, the Minister of NSTC, states that The Executive Yuan fully supports the development of Taiwan's space science and technology. NSTC will work together with diverse ministries and manufacturers in terms of policy, planning, law, and industry to lay out the blueprint for Taiwan's space development.
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[Sunny]NSTC Minister Wu Tsung-tsong (吳政忠) mentioned that the Executive Yuan strongly supports the development of Taiwan’s space technology. Besides, the NSTC will cooperate with other government ministries and industry circles on different aspects, to make the blueprint for the country's space development.
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[Wayne]...The restructuring shows Taiwan’s development into the new era in space exploration and the ambition to compete in the field of space.
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[Sarah] The minister of NSTC, told reporter that Executive Yuan will fully assist Taiwan's space development. NSTC will map out Taiwan's space development with departmants and the industry on policy, planning, and regulation. The establishment of TASA shoes that Taiwan is actively move into a new era of space development.
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[Daisy] Council Minister of NSTC Wu Tsung-tsong mentioned that the Executive Yuan will provide full support to the development of national space technology. NSTC will cooperate with all ministries and agencies from the aspects of policy, organization, legislation, and industry to draw up the blueprint for the advancement of the nation’s space industry.
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[Sofia] National Science and Technology Council (NSTC) Chairman Tsung-Tsong Wu said that the Executive Yuan fully supports the development of national space science and technology, and the NSTC joined hands with various ministries and industries to lay out Taiwan's space development blueprint in terms of policy, planning, the legal system, and industry. The establishment of the TASA demonstrates Taiwan's active entry into a new era of space developments and Taiwan's ambition to compete in the global space field.
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[Teika]
The NSTC minister Wu Cheng-Chung stated that Executive Yuan fully supports the development of Taiwan’s space science and technology, NSTC will collaborate with the government ministries and the sectors to depict a blueprint for space development from the aspect of policy, planning, legal system and industry.
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國家太空中心之中文名稱仍沿用，英文則由NSPO（National Space Organization）改為TASA（Taiwan Space Agency），除了提高縮寫可讀性，增加「臺灣」名稱亦提升於國際間識別度。

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[Jason]
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國家太空中心成立於1991年，最初為「國家太空計畫室籌備處」，旨在建立我國太空基礎設施並進行太空人才培育。

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[Wayne] Originally established as Taiwan Space Program Office in 1991, TASA aims to establish space infrastructures and cultivate astronautical talents.
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[Sunny] The Taiwan Space Agency was established on 1991, formerly known as the National Space Organization The purpose of the establishment is to build space infrastructure and rear potential astronauts.
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We are given the sequence $(x_n)$, where $x_0 = 1.2$ and $x_{n+1} = x_n + \sin(x_n)$ for $n\geq 0$. We want to determine whether this sequence converges, and if it does, what it converges to.

To show that the sequence converges, we will use the Monotone Convergence Theorem. First, note that $x_0 = 1.2 > 0$. Next, we will show that the sequence $(x_n)$ is increasing and bounded above.

To show that the sequence is increasing, we will use induction. For the base case, $x_0 = 1.2$. For the induction step, assume that $x_n < x_{n+1}$ for some $n \geq 0$. Then,

xn+1=xn+sin⁡(xn)>xnxn+1​=xn​+sin(xn​)>xn​

since $\sin(x_n) > 0$ for $x_n > 0$. Therefore, by induction, the sequence is increasing.

To show that the sequence is bounded above, we will use the fact that $\sin(x_n) \leq 1$ for all $n \geq 0$:

xn+1=xn+sin⁡(xn)≤xn+1xn+1​=xn​+sin(xn​)≤xn​+1

We can then show by induction that $x_n \leq n + 1.2$ for all $n \geq 0$. Therefore, the sequence is increasing and bounded above, so it must converge by the Monotone Convergence Theorem.

To find the limit of the sequence, let $L$ be the limit. Taking the limit of both sides of the recurrence relation, we get:

L=L+sin⁡(L)L=L+sin(L)

This implies that $\sin(L) = 0$, which means that $L = k\pi$ for some integer $k$. However, since $x_0 = 1.2 > 0$, we have $L > 0$. Therefore, $L = k\pi$ for some positive integer $k$. Since $x_n$ is increasing, we have $x_n \to L$ as $n \to \infty$. Thus, the sequence $(x_n)$ converges to the smallest positive solution of $\sin(x) = 0$, which is $x = \pi$. Therefore, $(x_n)$ converges to $\pi$.