确界原理证明单调有界原理

1from manim import *2import numpy as np3 4class MonotoneBoundedTheorem(Scene):5    def construct(self):6        # 配置参数7        x_range = [-1, 8, 1]8        y_range = [-0.5, 3.5, 0.5]9        10        # 创建坐标系11        axes = Axes(12            x_range=x_range,13            y_range=y_range,14            x_length=10,15            y_length=6,16            axis_config={17                "include_tip": True,18                "include_numbers": False19            }20        )21        22        # 创建标题23        title = Text("确界原理证明单调有界原理", font="SimSun").scale(0.8).to_edge(UP)24        25        # 设置场景26        self.play(27            Write(title, run_time=2),28            Create(axes, run_time=2)29        )30        31        # 创建单调递增数列 an = 2 - 1/n32        n_values = np.linspace(1, 8, 40)  # 在1到8之间创建40个点33        points = [(n, 2 - 1/n) for n in n_values]34        dots = VGroup(*[35            Dot(axes.c2p(x, y), color=BLUE, radius=0.04)36            for x, y in points37        ])38        39        # 显示数列说明40        sequence_text = VGroup(41            Text("单调递增数列", font="SimSun", color=BLUE),42            MathTex(r"\{a_n\}", color=BLUE)43        ).arrange(RIGHT,buff=0.1).scale(0.6)44        sequence_text.next_to(title, DOWN)45        46        # 显示数列47        self.play(Write(sequence_text), run_time=1.5)48        49        # 逐个显示点50        for dot in dots:51            self.play(Create(dot), run_time=0.2)52        53        # 显示性质2：有界性54        property2 = VGroup(55            Text("数列有上界必有上确界", font="SimSun"),56            MathTex(r"\exists L, \forall n, a_n \leq L")57        ).arrange(DOWN, aligned_edge=LEFT).scale(0.6).to_edge(LEFT, buff=0.5).shift(UP)58        59        self.play(Write(property2), run_time=2.5)60        61        # 显示上界L62        L = 2  # 上确界63        L_line = DashedLine(64            axes.c2p(0, L),65            axes.c2p(8, L),66            color=RED,67            dash_length=0.268        )69        L_text = Text("L（上确界）", font="SimSun", color=RED).scale(0.6)70        L_text.next_to(L_line.get_end(), LEFT)71        72        self.play(73            Create(L_line),74            Write(L_text),75            run_time=2.576        )77        78        # 显示证明过程79        proof = VGroup(80            Text("对任意ε>0，", font="SimSun"),81            Text("L-ε不是上界", font="SimSun", color=GREEN),82            Text("存在N，使得", font="SimSun"),83            MathTex(r"a_N > L-\varepsilon", color=YELLOW)84        ).arrange(DOWN, aligned_edge=LEFT).scale(0.6).to_edge(RIGHT, buff=0.5).shift(DOWN)85        86        self.play(Write(proof), run_time=3)87        88        # 演示ε-N语言89        epsilon = 0.290        epsilon_line = DashedLine(91            axes.c2p(0, L-epsilon),92            axes.c2p(8, L-epsilon),93            color=GREEN,94            dash_length=0.295        )96        epsilon_text = Text("L-ε", font="SimSun", color=GREEN).scale(0.6)97        epsilon_text.next_to(epsilon_line.get_end(), LEFT)98        99        self.play(100            Create(epsilon_line),101            Write(epsilon_text),102            run_time=2.5103        )104        105        # 标注N点106        N_index = next(i for i, p in enumerate(points) if p[1] > L-epsilon)107        N_point = points[N_index]108        N_dot = Dot(axes.c2p(N_point[0], N_point[1]), color=YELLOW, radius=0.1)109        N_text = Text("N", font="SimSun", color=YELLOW).scale(0.6)110        N_text.next_to(N_dot, DR)111        112        self.play(113            Create(N_dot),114            Write(N_text),115            run_time=2.5116        )117        118        # 显示性质1：单调性（移到N点显示之后）119        property1 = VGroup(120            Text("数列单调递增", font="SimSun"),121            MathTex(r"\forall n>N, a_n \geq a_N > L-\varepsilon", color=BLUE)122        ).arrange(DOWN, aligned_edge=LEFT).scale(0.6).to_edge(LEFT, buff=0.5)123        124        property1.next_to(property2, DOWN, buff=0.5, aligned_edge=LEFT)125        126        self.play(Write(property1), run_time=2.5)127        128        # 标注N之后的点129        points_after_N = [(x, y) for x, y in points if x > N_point[0]]130        highlight_dots = VGroup(*[131            Dot(axes.c2p(x, y), color=YELLOW, radius=0.08, fill_opacity=0.8)132            for x, y in points_after_N133        ])134        135        # 显示动画136        self.play(137            *[dot.animate.set_color(YELLOW) for dot in highlight_dots],138            run_time=3139        )140        141        # 突出显示两条边界线142        self.play(143            epsilon_line.animate.set_color(YELLOW).set_stroke(width=3),144            L_line.animate.set_color(YELLOW).set_stroke(width=3),145            run_time=2.5146        )147        148        # 显示不等式链149        inequality_chain = VGroup(150            MathTex("L-\\varepsilon", color=GREEN),151            MathTex("<"),152            MathTex("a_n", color=BLUE),153            MathTex("\\leq"),154            MathTex("L", color=RED)155        ).arrange(RIGHT, buff=0.2).scale(0.8)156        157        inequality_chain.next_to(axes, DOWN, buff=0.15)158        159        self.play(Write(inequality_chain), run_time=3.5)160        161        # 显示结论162        conclusion = Text(163            "数列必定收敛于其上确界L",164            font="SimSun"165        ).scale(0.6).next_to(inequality_chain, DOWN, buff=0.15)166        167        self.play(Write(conclusion), run_time=3.5)168        169        self.wait(6)170 171def main():172    import os173    os.system("manim -pql monotone_bounded_theorem.py MonotoneBoundedTheorem")174 175if __name__ == "__main__":176    main()