用柯西收敛原理的反命题证明极限不存在

1from manim import *2import numpy as np3 4# 中文 LaTeX 模板与中文主字体5config.tex_template = TexTemplateLibrary.ctex6config.tex_template.add_to_preamble(r"\setCJKmainfont{STSong}")7 8class CauchyLimitDoesNotExist(Scene):9    def construct(self):10        # 统一字体大小11        Text.set_default(font="STSong", font_size=36)12        MathTex.set_default(font_size=34)13 14        # 标题15        title = Text("用柯西收敛原理的反命题证明极限不存在", color=YELLOW).to_edge(UP)16        self.play(FadeIn(title, shift=DOWN), run_time=1.8)17        self.wait(1.2)18 19        # 数列与反命题：放在标题正下，反命题在数列右侧；数学符号使用 MathTex20        seq_def = VGroup(21            Text("数列：", font_size=30),22            MathTex(r"a_n = (-1)^n", font_size=34)23        ).arrange(RIGHT, buff=0.2)24 25        prop_label = Text("柯西收敛原理反命题：", font_size=30, color=GREEN)26        prop_math = MathTex(r"\exists\,\varepsilon>0,\ \forall\,N,\ \exists\,m,n>N:\ |a_n-a_m|\ge \varepsilon", font_size=30)27        prop = VGroup(prop_label, prop_math).arrange(DOWN, aligned_edge=LEFT, buff=0.15)28 29        header_row = VGroup(seq_def, prop).arrange(RIGHT, buff=0.8)30        header_row.next_to(title, DOWN, buff=0.45).to_edge(LEFT, buff=0.7)31        self.play(Write(seq_def), run_time=1.4)32        self.play(Write(prop_label), run_time=1.4)33        self.play(Write(prop_math), run_time=1.4)34        self.wait(1.0)35 36        # 坐标系与点列（整体上移）37        axes = Axes(38            x_range=[1, 20, 1], y_range=[-2, 2, 1],39            x_length=10, y_length=4,40            axis_config={"include_tip": True}41        )42        x_label = axes.get_x_axis_label(MathTex("n", font_size=28))43        y_label = axes.get_y_axis_label(MathTex("a_n", font_size=28))44        axes_group = VGroup(axes, x_label, y_label)45        axes_group.to_edge(DOWN).shift(UP*0.8)46        self.play(Create(axes), FadeIn(x_label, shift=DOWN), FadeIn(y_label, shift=LEFT), run_time=2.0)47 48        dots = VGroup()49        for n in range(1, 21):50            y = (-1)**n51            d = Dot(axes.c2p(n, y), radius=0.06, color=BLUE)52            dots.add(d)53        self.play(LaggedStartMap(FadeIn, dots, lag_ratio=0.07, run_time=3.0))54        self.wait(1.2)55 56        # 取 ε=1 放在图下方（数学符号 MathTex）57        eps_grp = VGroup(58            Text("取：", font_size=30), MathTex(r"\varepsilon = 1", font_size=32, color=ORANGE)59        ).arrange(RIGHT, buff=0.2)60        eps_grp.next_to(axes, DOWN, buff=0.45).to_edge(LEFT, buff=0.7)61        self.play(Write(eps_grp), run_time=1.2)62        self.wait(0.9)63 64        # 演示给定 N，明确说明“对该 N 可取 m=N+1,n=N+2”，所有含符号处使用 MathTex65        def show_case(N: int):66            m, n = N+1, N+267            y_m = (-1)**m68            y_n = (-1)**n69 70            # 在图中明确演示 N71            vline_N = DashedLine(axes.c2p(N, 1.4), axes.c2p(N, -1.4), color=WHITE, stroke_width=2, dashed_ratio=0.55)72            markN = MathTex(fr"N={N}", font_size=26, color=WHITE).next_to(axes.c2p(N, 0), DOWN, buff=0.2)73            Nline = Arrow(axes.c2p(N, -0.15), axes.c2p(N, 0.0), buff=0, color=WHITE, max_tip_length_to_length_ratio=0.18, stroke_width=2)74 75            # 高亮两点与标签76            dm = Dot(axes.c2p(m, y_m), color=RED, radius=0.09)77            dn = Dot(axes.c2p(n, y_n), color=RED, radius=0.09)78            lab_m = MathTex(fr"a_{{{m}}}", font_size=26, color=RED).next_to(dm, UP if y_m>0 else DOWN, buff=0.1)79            lab_n = MathTex(fr"a_{{{n}}}", font_size=26, color=RED).next_to(dn, UP if y_n>0 else DOWN, buff=0.1)80 81            # 辅助竖线、距离与说明82            vline_m = DashedLine(axes.c2p(m, 1.2), axes.c2p(m, -1.2), color=GREY_B, stroke_width=2)83            vline_n = DashedLine(axes.c2p(n, 1.2), axes.c2p(n, -1.2), color=GREY_B, stroke_width=2)84            seg = Line(axes.c2p(m, y_m), axes.c2p(n, y_n), color=YELLOW, stroke_width=4)85            dist_brace = BraceBetweenPoints(axes.c2p(m, y_m), axes.c2p(n, y_n), direction=RIGHT if m<n else LEFT)86            dist_label = MathTex(r"|a_n-a_m| = 2 > 1 = \varepsilon", font_size=28, color=YELLOW).next_to(dist_brace, RIGHT, buff=0.15).shift(UP*0.2)87 88            # 图下说明（使用 MathTex 拼接，确保符号为数学格式，放在epsilon=1右侧）89            expl = VGroup(90                Text("当 ", font_size=28, color=GREEN),91                MathTex(fr"N={N}", font_size=28, color=GREEN),92                Text(" 时：取 ", font_size=28, color=GREEN),93                MathTex(fr"m={m},\ n={n}", font_size=28, color=GREEN),94                Text("，有 ", font_size=28, color=GREEN),95                MathTex(r"|a_n-a_m|=2\ge 1", font_size=28, color=GREEN)96            ).arrange(RIGHT, buff=0.15)97            expl.next_to(eps_grp, RIGHT, buff=0.4)98 99            self.play(Create(vline_N), FadeIn(markN), Create(Nline), run_time=1.2)100            self.play(Create(vline_m), Create(vline_n), FadeIn(dm), FadeIn(dn), Write(lab_m), Write(lab_n), run_time=1.8)101            self.play(Create(seg), run_time=1.2)102            self.play(GrowFromCenter(dist_brace), FadeIn(dist_label), run_time=1.2)103            self.play(FadeIn(expl), run_time=1.1)104            self.wait(1.2)105 106            # 清理为下一轮（保留坐标轴与蓝点、上方文字）107            self.play(108                FadeOut(expl), FadeOut(vline_m), FadeOut(vline_n), FadeOut(seg),109                FadeOut(dist_brace), FadeOut(dist_label),110                FadeOut(dm), FadeOut(dn), FadeOut(lab_m), FadeOut(lab_n),111                FadeOut(markN), FadeOut(Nline), FadeOut(vline_N),112                run_time=1.2113            )114 115        for N in [3, 6, 11, 15]:116            show_case(N)117 118        # 在N=15演示完后，说明对于任意N的通用情况119        general_explanation = VGroup(120            Text("对于任意 ", font_size=28, color=BLUE),121            MathTex("N", font_size=28, color=BLUE),122            Text("，取 ", font_size=28, color=BLUE),123            MathTex("m=N+1", font_size=28, color=BLUE),124            Text("，", font_size=28, color=BLUE),125            MathTex("n=N+2", font_size=28, color=BLUE),126            Text("，有 ", font_size=28, color=BLUE),127            MathTex(r"|a_n-a_m| = 2 \ge 1 = \varepsilon", font_size=24, color=BLUE)128        ).arrange(RIGHT, buff=0.1)129        general_explanation.next_to(eps_grp, RIGHT, buff=0.4)130        131        self.play(Write(general_explanation), run_time=2.0)132        self.wait(1.5)133 134        # 总解释：不同 N 都能找到这样的 m,n（图下；数学符号 MathTex）135        general_note = VGroup(136            Text("因此对任意 ", font_size=28, color=ORANGE),137            MathTex("N", font_size=28, color=ORANGE),138            Text("，总能找到 ", font_size=28, color=ORANGE),139            MathTex(r"m=N+1,\ n=N+2", font_size=28, color=ORANGE),140            Text(" 使 ", font_size=28, color=ORANGE),141            MathTex(r"|a_n-a_m|\ge 1", font_size=28, color=ORANGE),142            Text("。", font_size=28, color=ORANGE)143        ).arrange(RIGHT, buff=0.12).next_to(eps_grp, DOWN, buff=0.6).to_edge(LEFT, buff=0.7)144        self.play(Write(general_note), run_time=1.4)145        self.wait(1.0)146 147        # 总结（放在"对于任意N"上面，一行显示，左右居中）148        summary = VGroup(149            Text("结论：", font_size=30, color=YELLOW),150            MathTex(r"(a_n) \text{ 不是柯西列 } \Rightarrow \text{ 极限不存在}", font_size=32, color=YELLOW)151        ).arrange(RIGHT, buff=0.3).next_to(general_explanation, UP, buff=0.4).move_to(ORIGIN + UP * (general_explanation.get_center()[1] + 0.4))152        box = SurroundingRectangle(summary, color=YELLOW, buff=0.28)153        self.play(Write(summary), run_time=1.5)154        self.play(Create(box), run_time=1.2)155        self.wait(2.0)