Let \((x_{n})\, n\in\mathbf{N}\) be a convergent sequence of positive real valued numbers. Which of the following statements is true ?
The limit of every subsequence of the sequence is non-negative
Limits preserve non-strict inequalities: if \(x_{n}\gt 0\) for every \(n\) and \(x_{n}\to L\), then \(L\geq 0\). Strict positivity of the terms need not survive in the limit.
So the limit can never be negative. If \(L\lt 0\), then for \(\varepsilon=|L|/2\) all large terms would satisfy \(x_{n}\lt L/2\lt 0\), contradicting positivity.
The limit need not be positive: \(x_{n}=\frac{1}{n}\) has positive terms but converges to \(0\).
For the same sequence every subsequence also converges to \(0\), so there is no subsequence with a positive limit; a subsequence with positive limit is therefore not guaranteed.
Every subsequence of a convergent sequence converges to the same limit \(L\), and \(L\geq 0\), so every subsequential limit is non-negative.
Check: for \(x_{n}=\frac{1}{n}\) the subsequence \(x_{2n}=\frac{1}{2n}\) tends to \(0\), which is non-negative but not positive, exactly as the reasoning predicts.
Hence, The limit of every subsequence of the sequence is non-negative.
Similar Questions
Let \(X\) be a connected metric space. Then which of the following is true ?
The series :
\(\frac{2}{1} + \frac{3}{4} + \frac{4}{9} + \ldots + \frac{n+1}{n^2} + \ldots\)
Let \(X, Y\) be metric spaces. Suppose \(f, g : X \to Y\) are continuous functions and \(D = \{x \mid f(x) = g(x)\}\). Then \(f \equiv g\) if \(D\) is :
Which of the following statements is not true in any metric space ?
Suppose that \(\{f_n\}\) is a sequence of functions defined by :
\(f_n(x) = \frac{x^n}{1 + x^n}, \ 0 \le x \le 1.\)
Then, which of the following statements is true ?
\(S_1 : f_n \to f \equiv 0\) uniformly on \([0, 1]\)
\(S_2 : \sup_{0 \le x \le \frac{1}{2}} f_n(x) \to F(x) \equiv 0\) on \([0, \frac{1}{2}]\)
\(S_3 : f_n \to f = 0\) uniformly on \([0, \frac{1}{2}]\)
Suppose that \(f\) is a function on \(\mathbb{R}^2\) defined by :
\(f(x, y) = \begin{cases} \frac{xy}{x^2 + y^2}, & (x, y) \neq (0, 0) \\ 0, & (x, y) = (0, 0) \end{cases}\)
Then, which of the following statements is/are true ?
\(S_1\) : \(f\) is continuous at \((0, 0)\).
\(S_2\) : Both partial derivatives exist at \((0, 0)\).
For any two sequences \(\{x_n\}\) and \(\{y_n\}\), it is known that,
\(\liminf x_n + \liminf y_n \le \liminf (x_n + y_n) \le \limsup (x_n + y_n) \le \limsup x_n + \limsup y_n \quad (*)\)
Suppose that \(x_n = \begin{cases} 0 & \text{if } n \text{ is even} \\ 1 & \text{if } n \text{ is odd} \end{cases}\) and \(y_n = 1 - x_n\), for all \(n \ge 1\).
Then, which of the following statements is/are true for \(\{x_n\}\) and \(\{y_n\}\) ?
\(S_1\) : strict inequalities hold in \((*)\) above.
\(S_2\) : \(\{x_n + y_n\}\) converges, while \(\{x_n\}\) and \(\{y_n\}\) do not.
Suppose that \(f\) is a function defined by :
\(f(x) = \frac{x}{3 - 2x}, \ 0 < x < 1\)
If \(g(x) = f''(x) + f'(x) - [f'(x)]^2\), then \(g(1)\) is equal to :
Consider the function \(f\) defined by :
\(f(x) = \begin{cases} \frac{\lambda \cos(x)}{\pi/2 - x}, & x \neq \pi/2 \\ 1, & x = \pi/2 \end{cases}\)
What should be the value of \(\lambda\) so that \(f\) would be continuous at \(x = \pi/2\) ?
Assuming \((X - I)^{-1}\) exists, what will be \(I + X + X^2 + \ldots + X^{n-1}\) ?
Important Questions from Mixed Topic (MH-SET)
.................... is a client server application that identifies each host on the internet with a unique user-friendly name.
Every OS has a ................ or .......... which primarily resides in the main memory of the computer to perform some of the basic functions of the OS.
(i) Kernel
(ii) Interface
(iii) Nucleus
(iv) Compiler
Codes:
Which of the following are referred to as low level language?
(i) Machine language
(ii) Assembly language
(iii) High level language
(iv) Natural language
Codes:
Match the following:
List I List II (a) New York Public Library (i) Shelv AR (b) Miami University Augmented Research Group (ii) Smart library (c) Oula University Library (iii) Wolf Walk (d) North Carolina State University (iv) Find the future Codes:
Which part of the QR code represents the encoded data?
