Let \((X_{n})\) be a sequence of real numbers and \(x\in\mathbf{R}\). Then which of the following is true ?
If \(\lim\limits_{n\to\infty} x_{n}\lt x\), then \(x_{n}\lt x\) for infinitely many numbers \(n\)
Let \(L=\lim\limits_{n\to\infty} x_{n}\) with \(L\lt x\). Take \(\varepsilon=\frac{x-L}{2}\gt 0\); by the definition of the limit there is \(N\) with \(|x_{n}-L|\lt\varepsilon\) for all \(n\geq N\).
For those \(n\), \(x_{n}\lt L+\varepsilon=\frac{L+x}{2}\lt x\). So \(x_{n}\lt x\) for every \(n\geq N\), which is certainly infinitely many indices.
A limit only controls the tail, not every term: \(x_{n}=\frac{1}{n}\) has limit \(0\) and \(0\leq 0\), but \(x_{1}=1\gt 0\), so the claim for all \(n\) fails.
A different limit does not force terms on both sides of \(x\): the constant sequence \(x_{n}=1\) with \(x=0\) has limit \(1\neq 0\), yet no term is below \(0\).
A divergent sequence can still have convergent subsequences: \(x_{n}=(-1)^{n}\) has no limit, but \(x_{2n}=1\) converges to \(x=1\).
Check: for \(x_{n}=\frac{1}{n}\) and \(x=\frac{1}{2}\), the limit \(0\) is below \(x\), and indeed \(x_{n}\lt\frac{1}{2}\) for every \(n\geq 3\), infinitely many indices.
Hence, If \(\lim\limits_{n\to\infty} x_{n}\lt x\), then \(x_{n}\lt x\) for infinitely many numbers \(n\).
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?
