hoelzl@57025: theory NthRoot_Limits
hoelzl@57025:   imports Complex_Main "~~/src/HOL/Number_Theory/Binomial"
hoelzl@57025: begin
hoelzl@57025: 
hoelzl@57025: text {*
hoelzl@57025: 
hoelzl@57025: This does not fit into @{text Complex_Main}, as it depends on @{text Binomial}
hoelzl@57025: 
hoelzl@57025: *}
hoelzl@57025: 
hoelzl@57025: lemma LIMSEQ_root: "(\<lambda>n. root n n) ----> 1"
hoelzl@57025: proof -
hoelzl@57025:   def x \<equiv> "\<lambda>n. root n n - 1"
hoelzl@57025:   have "x ----> sqrt 0"
hoelzl@57025:   proof (rule tendsto_sandwich[OF _ _ tendsto_const])
hoelzl@57025:     show "(\<lambda>x. sqrt (2 / x)) ----> sqrt 0"
hoelzl@57025:       by (intro tendsto_intros tendsto_divide_0[OF tendsto_const] filterlim_mono[OF filterlim_real_sequentially])
hoelzl@57025:          (simp_all add: at_infinity_eq_at_top_bot)
hoelzl@57025:     { fix n :: nat assume "2 < n"
hoelzl@57025:       have "1 + (real (n - 1) * n) / 2 * x n^2 = 1 + of_nat (n choose 2) * x n^2"
hoelzl@57025:         using `2 < n` unfolding gbinomial_def binomial_gbinomial
hoelzl@57025:         by (simp add: atLeast0AtMost atMost_Suc field_simps real_of_nat_diff numeral_2_eq_2 real_eq_of_nat[symmetric])
hoelzl@57025:       also have "\<dots> \<le> (\<Sum>k\<in>{0, 2}. of_nat (n choose k) * x n^k)"
hoelzl@57025:         by (simp add: x_def)
hoelzl@57025:       also have "\<dots> \<le> (\<Sum>k=0..n. of_nat (n choose k) * x n^k)"
hoelzl@57025:         using `2 < n` by (intro setsum_mono2) (auto intro!: mult_nonneg_nonneg zero_le_power simp: x_def le_diff_eq)
hoelzl@57025:       also have "\<dots> = (x n + 1) ^ n"
hoelzl@57025:         by (simp add: binomial_ring)
hoelzl@57025:       also have "\<dots> = n"
hoelzl@57025:         using `2 < n` by (simp add: x_def)
hoelzl@57025:       finally have "real (n - 1) * (real n / 2 * (x n)\<^sup>2) \<le> real (n - 1) * 1"
hoelzl@57025:         by simp
hoelzl@57025:       then have "(x n)\<^sup>2 \<le> 2 / real n"
hoelzl@57025:         using `2 < n` unfolding mult_le_cancel_left by (simp add: field_simps)
hoelzl@57025:       from real_sqrt_le_mono[OF this] have "x n \<le> sqrt (2 / real n)"
hoelzl@57025:         by simp }
hoelzl@57025:     then show "eventually (\<lambda>n. x n \<le> sqrt (2 / real n)) sequentially"
hoelzl@57025:       by (auto intro!: exI[of _ 3] simp: eventually_sequentially)
hoelzl@57025:     show "eventually (\<lambda>n. sqrt 0 \<le> x n) sequentially"
hoelzl@57025:       by (auto intro!: exI[of _ 1] simp: eventually_sequentially le_diff_eq x_def)
hoelzl@57025:   qed
hoelzl@57025:   from tendsto_add[OF this tendsto_const[of 1]] show ?thesis
hoelzl@57025:     by (simp add: x_def)
hoelzl@57025: qed
hoelzl@57025: 
hoelzl@57025: lemma LIMSEQ_root_const:
hoelzl@57025:   assumes "0 < c"
hoelzl@57025:   shows "(\<lambda>n. root n c) ----> 1"
hoelzl@57025: proof -
hoelzl@57025:   { fix c :: real assume "1 \<le> c"
hoelzl@57025:     def x \<equiv> "\<lambda>n. root n c - 1"
hoelzl@57025:     have "x ----> 0"
hoelzl@57025:     proof (rule tendsto_sandwich[OF _ _ tendsto_const])
hoelzl@57025:       show "(\<lambda>n. c / n) ----> 0"
hoelzl@57025:         by (intro tendsto_divide_0[OF tendsto_const] filterlim_mono[OF filterlim_real_sequentially])
hoelzl@57025:            (simp_all add: at_infinity_eq_at_top_bot)
hoelzl@57025:       { fix n :: nat assume "1 < n"
hoelzl@57025:         have "1 + x n * n = 1 + of_nat (n choose 1) * x n^1"
hoelzl@57025:           using `1 < n` unfolding gbinomial_def binomial_gbinomial by (simp add: real_eq_of_nat[symmetric])
hoelzl@57025:         also have "\<dots> \<le> (\<Sum>k\<in>{0, 1}. of_nat (n choose k) * x n^k)"
hoelzl@57025:           by (simp add: x_def)
hoelzl@57025:         also have "\<dots> \<le> (\<Sum>k=0..n. of_nat (n choose k) * x n^k)"
hoelzl@57025:           using `1 < n` `1 \<le> c` by (intro setsum_mono2) (auto intro!: mult_nonneg_nonneg zero_le_power simp: x_def le_diff_eq)
hoelzl@57025:         also have "\<dots> = (x n + 1) ^ n"
hoelzl@57025:           by (simp add: binomial_ring)
hoelzl@57025:         also have "\<dots> = c"
hoelzl@57025:           using `1 < n` `1 \<le> c` by (simp add: x_def)
hoelzl@57025:         finally have "x n \<le> c / n"
hoelzl@57025:           using `1 \<le> c` `1 < n` by (simp add: field_simps) }
hoelzl@57025:       then show "eventually (\<lambda>n. x n \<le> c / n) sequentially"
hoelzl@57025:         by (auto intro!: exI[of _ 3] simp: eventually_sequentially)
hoelzl@57025:       show "eventually (\<lambda>n. 0 \<le> x n) sequentially"
hoelzl@57025:         using `1 \<le> c` by (auto intro!: exI[of _ 1] simp: eventually_sequentially le_diff_eq x_def)
hoelzl@57025:     qed
hoelzl@57025:     from tendsto_add[OF this tendsto_const[of 1]] have "(\<lambda>n. root n c) ----> 1"
hoelzl@57025:       by (simp add: x_def) }
hoelzl@57025:   note ge_1 = this
hoelzl@57025: 
hoelzl@57025:   show ?thesis
hoelzl@57025:   proof cases
hoelzl@57025:     assume "1 \<le> c" with ge_1 show ?thesis by blast
hoelzl@57025:   next
hoelzl@57025:     assume "\<not> 1 \<le> c"
hoelzl@57025:     with `0 < c` have "1 \<le> 1 / c"
hoelzl@57025:       by simp
hoelzl@57025:     then have "(\<lambda>n. 1 / root n (1 / c)) ----> 1 / 1"
hoelzl@57025:       by (intro tendsto_divide tendsto_const ge_1 `1 \<le> 1 / c` one_neq_zero)
hoelzl@57025:     then show ?thesis
hoelzl@57025:       by (rule filterlim_cong[THEN iffD1, rotated 3])
hoelzl@57025:          (auto intro!: exI[of _ 1] simp: eventually_sequentially real_root_divide)
hoelzl@57025:   qed
hoelzl@57025: qed
hoelzl@57025: 
hoelzl@57025: end