- #1
yucheng
- 232
- 57
####
If ##b \leq x_n \leq c## for all but a finite number of n, show that ##b \leq \operatorname{lim inf}_{n \to \infty} x_n## and ##\operatorname{lim sup}_{n \to \infty} x_n \leq c_n##
(Buck, Advanced Calculus, Section 1.6, Exercise 24)
Let ##\beta =\operatorname{lim inf}_{n \to \infty} x_n## and ##\alpha = \operatorname{lim sup}_{n \to \infty} x_n##. Let ##\varepsilon## be any number greater than 0. Since ##\beta## and ##\alpha## are limit points, there exists subsequences of integers ##\{n_k\}## and ##\{n_i\}##, both infinite, such that ##|x_{n_k}-\beta| \leq \varepsilon## and ##|x_{n_i}-\alpha| \leq \varepsilon## Then, ##-\varepsilon \geq x_{n_k}-\beta \leq \varepsilon## and ##-\varepsilon \geq x_{n_i}-\alpha \leq \varepsilon##. From this, we get $$b \leq x_{n_k} \leq \beta + \varepsilon$$ and $$\alpha - \varepsilon \leq x_{n_i} \leq c$$ Since ##\varepsilon## is arbitrarily small, is it true that the inequalities above become ##b \leq \beta## and ##\alpha \leq c##
If ##b \leq x_n \leq c## for all but a finite number of n, show that ##b \leq \operatorname{lim inf}_{n \to \infty} x_n## and ##\operatorname{lim sup}_{n \to \infty} x_n \leq c_n##
(Buck, Advanced Calculus, Section 1.6, Exercise 24)
Let ##\beta =\operatorname{lim inf}_{n \to \infty} x_n## and ##\alpha = \operatorname{lim sup}_{n \to \infty} x_n##. Let ##\varepsilon## be any number greater than 0. Since ##\beta## and ##\alpha## are limit points, there exists subsequences of integers ##\{n_k\}## and ##\{n_i\}##, both infinite, such that ##|x_{n_k}-\beta| \leq \varepsilon## and ##|x_{n_i}-\alpha| \leq \varepsilon## Then, ##-\varepsilon \geq x_{n_k}-\beta \leq \varepsilon## and ##-\varepsilon \geq x_{n_i}-\alpha \leq \varepsilon##. From this, we get $$b \leq x_{n_k} \leq \beta + \varepsilon$$ and $$\alpha - \varepsilon \leq x_{n_i} \leq c$$ Since ##\varepsilon## is arbitrarily small, is it true that the inequalities above become ##b \leq \beta## and ##\alpha \leq c##
Last edited: