Real Analysis

Question 15 (CSIR NET June 2019)

Let $g_n(x) = \frac{nx}{1+n^2x^2}$, $x \in [0, \infty)$, let $n \to \infty$, then- (a) $g_n \to 0$ pointwise but not uniform…

Question 14 (CSIR NET June 2019)

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous and one-one function, then- (a) $f$ is onto (b) $f$ is either strictly i…

Question 13 (Real Analysis)

Let $f:(0,1)\rightarrow \mathbb{R}$ is injective function, then choose the correct options- (a) Range of $f$ must contains r…

Question 12 (Countability of a Set)

Let $A=[0,1]\cap \mathbb{Q}$ and suppose $B_n=\{x \in A | x=\frac{p}{q},  q \leq n, q \in \mathbb{Z}^+\}$ for all $n \in \mathbb…

Cantor's Theorem

Statement of the theorem Let $A$ be a non-empty set, then there does not exist any onto function $f$ from $A$ to $P(A)$. Proof o…

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