src/HOL/Library/NthRoot_Limits.thy
 author hoelzl Tue May 20 19:24:39 2014 +0200 (2014-05-20) changeset 57025 e7fd64f82876 child 59667 651ea265d568 permissions -rw-r--r--
1 theory NthRoot_Limits
2   imports Complex_Main "~~/src/HOL/Number_Theory/Binomial"
3 begin
5 text {*
7 This does not fit into @{text Complex_Main}, as it depends on @{text Binomial}
9 *}
11 lemma LIMSEQ_root: "(\<lambda>n. root n n) ----> 1"
12 proof -
13   def x \<equiv> "\<lambda>n. root n n - 1"
14   have "x ----> sqrt 0"
15   proof (rule tendsto_sandwich[OF _ _ tendsto_const])
16     show "(\<lambda>x. sqrt (2 / x)) ----> sqrt 0"
17       by (intro tendsto_intros tendsto_divide_0[OF tendsto_const] filterlim_mono[OF filterlim_real_sequentially])
18          (simp_all add: at_infinity_eq_at_top_bot)
19     { fix n :: nat assume "2 < n"
20       have "1 + (real (n - 1) * n) / 2 * x n^2 = 1 + of_nat (n choose 2) * x n^2"
21         using `2 < n` unfolding gbinomial_def binomial_gbinomial
22         by (simp add: atLeast0AtMost atMost_Suc field_simps real_of_nat_diff numeral_2_eq_2 real_eq_of_nat[symmetric])
23       also have "\<dots> \<le> (\<Sum>k\<in>{0, 2}. of_nat (n choose k) * x n^k)"
24         by (simp add: x_def)
25       also have "\<dots> \<le> (\<Sum>k=0..n. of_nat (n choose k) * x n^k)"
26         using `2 < n` by (intro setsum_mono2) (auto intro!: mult_nonneg_nonneg zero_le_power simp: x_def le_diff_eq)
27       also have "\<dots> = (x n + 1) ^ n"
28         by (simp add: binomial_ring)
29       also have "\<dots> = n"
30         using `2 < n` by (simp add: x_def)
31       finally have "real (n - 1) * (real n / 2 * (x n)\<^sup>2) \<le> real (n - 1) * 1"
32         by simp
33       then have "(x n)\<^sup>2 \<le> 2 / real n"
34         using `2 < n` unfolding mult_le_cancel_left by (simp add: field_simps)
35       from real_sqrt_le_mono[OF this] have "x n \<le> sqrt (2 / real n)"
36         by simp }
37     then show "eventually (\<lambda>n. x n \<le> sqrt (2 / real n)) sequentially"
38       by (auto intro!: exI[of _ 3] simp: eventually_sequentially)
39     show "eventually (\<lambda>n. sqrt 0 \<le> x n) sequentially"
40       by (auto intro!: exI[of _ 1] simp: eventually_sequentially le_diff_eq x_def)
41   qed
42   from tendsto_add[OF this tendsto_const[of 1]] show ?thesis
43     by (simp add: x_def)
44 qed
46 lemma LIMSEQ_root_const:
47   assumes "0 < c"
48   shows "(\<lambda>n. root n c) ----> 1"
49 proof -
50   { fix c :: real assume "1 \<le> c"
51     def x \<equiv> "\<lambda>n. root n c - 1"
52     have "x ----> 0"
53     proof (rule tendsto_sandwich[OF _ _ tendsto_const])
54       show "(\<lambda>n. c / n) ----> 0"
55         by (intro tendsto_divide_0[OF tendsto_const] filterlim_mono[OF filterlim_real_sequentially])
56            (simp_all add: at_infinity_eq_at_top_bot)
57       { fix n :: nat assume "1 < n"
58         have "1 + x n * n = 1 + of_nat (n choose 1) * x n^1"
59           using `1 < n` unfolding gbinomial_def binomial_gbinomial by (simp add: real_eq_of_nat[symmetric])
60         also have "\<dots> \<le> (\<Sum>k\<in>{0, 1}. of_nat (n choose k) * x n^k)"
61           by (simp add: x_def)
62         also have "\<dots> \<le> (\<Sum>k=0..n. of_nat (n choose k) * x n^k)"
63           using `1 < n` `1 \<le> c` by (intro setsum_mono2) (auto intro!: mult_nonneg_nonneg zero_le_power simp: x_def le_diff_eq)
64         also have "\<dots> = (x n + 1) ^ n"
65           by (simp add: binomial_ring)
66         also have "\<dots> = c"
67           using `1 < n` `1 \<le> c` by (simp add: x_def)
68         finally have "x n \<le> c / n"
69           using `1 \<le> c` `1 < n` by (simp add: field_simps) }
70       then show "eventually (\<lambda>n. x n \<le> c / n) sequentially"
71         by (auto intro!: exI[of _ 3] simp: eventually_sequentially)
72       show "eventually (\<lambda>n. 0 \<le> x n) sequentially"
73         using `1 \<le> c` by (auto intro!: exI[of _ 1] simp: eventually_sequentially le_diff_eq x_def)
74     qed
75     from tendsto_add[OF this tendsto_const[of 1]] have "(\<lambda>n. root n c) ----> 1"
76       by (simp add: x_def) }
77   note ge_1 = this
79   show ?thesis
80   proof cases
81     assume "1 \<le> c" with ge_1 show ?thesis by blast
82   next
83     assume "\<not> 1 \<le> c"
84     with `0 < c` have "1 \<le> 1 / c"
85       by simp
86     then have "(\<lambda>n. 1 / root n (1 / c)) ----> 1 / 1"
87       by (intro tendsto_divide tendsto_const ge_1 `1 \<le> 1 / c` one_neq_zero)
88     then show ?thesis
89       by (rule filterlim_cong[THEN iffD1, rotated 3])
90          (auto intro!: exI[of _ 1] simp: eventually_sequentially real_root_divide)
91   qed
92 qed
94 end