New posts in Real Analysis

$\lim\limits_{n\to\infty}n\big(\sum_{k=1}^n\frac{k^2}{n^3+kn}-\frac{1}{3}\big)$?

If for every sequence $(x_n)$, $x_n \to x_0 \implies (f(x_n))$ is a Cauchy sequence, then is $f$ continuous?

Sign of the first derivative

$\lim_{ x \to 0 } \frac{e-(1+\arctan x)^{\frac{1}{x}}}{x}$

For any $S \subseteq \mathbf{R}$, if $\partial S$ denotes the boundary of $S$, prove that $\partial(\partial S) \subseteq \partial S$.

The proofs of limit laws and derivative rules appear to tacitly assume that the limit exists in the first place

Formalizing composition of sequences

Infinite Rational numbers add up to give irrational numbers.

Given $f(i, j) = f(i − 1, j − 1) + f(i, j − 1)$, find the value of $f(1009, 2019)$

$g(x) = \frac{f'(x)}{f(x)}$ not bounded