Question

If $f_n\to f$ in $L^p$ and $g_n$ is uniformly bounded and $g_n\to g$ a.e. then $g_nf_n\to gf$ in $L^p$.

score 415 · Answer 1 · 2026-05-10 17:39:15

415

Views

If $f_n\to f$ in $L^p$ and $g_n$ is uniformly bounded and $g_n\to g$ a.e. then $g_nf_n\to gf$ in $L^p$.

Published on 10 May 2026 - 17:39

score 14.4k · Answer 2 · 2026-05-10 13:47:37

14.4k

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My Proof: Every convergent sequence is a Cauchy sequence.

Published on 10 May 2026 - 13:47

score 53 · Answer 3 · 2026-05-10 22:02:47

Let $f:[a,b]\to \mathbb{R}$ is integrable on $[a,b]$. Show that $f_1:[a,b]\to \mathbb{R}$ such that $f_1(x)=\sup\{f(x), 0\}$, is integrable.

score 142 · Answer 4 · 2026-04-20 11:10:21

142

Views

How to prove this specific function has limit either $\infty$ or 0?

Published on 20 Apr 2026 - 11:10

score 1.2k · Answer 5 · 2026-04-14 20:51:40

1.2k

Views

Whether the indefinite integral of a differentiable function is differentiable or not

Published on 14 Apr 2026 - 20:51

score 183 · Answer 6 · 2026-05-10 21:45:18

183

Views

Chacking integrability of a piecewise function

Published on 10 May 2026 - 21:45

score 125 · Answer 7 · 2026-05-14 19:39:13

125

Views

Prove or disprove that $(b-a)^2(\log(b)-\log(a))^{-2}\ge ab$ for all $a,b>0$

Published on 14 May 2026 - 19:39

score 43 · Answer 8 · 2026-04-12 05:33:39

43

Views

Finding a function that solves the condition

Published on 12 Apr 2026 - 5:33

score 163 · Answer 9 · 2026-04-24 03:51:53

163

Views

proof $\lim x_n \le \lim y_n$ if $x_n \le y_n$ (clarification)

Published on 24 Apr 2026 - 3:51

score 2.1k · Answer 10 · 2026-04-14 16:48:01

2.1k

Views

Monotone convergence theorem implies Nested Interval property

Published on 14 Apr 2026 - 16:48

score 183 · Answer 11 · 2026-05-11 01:25:07

183

Views

Exponentials inequality: $|e^x + e^y| \leq |x - y|$ for $x,y<0$.

Published on 11 May 2026 - 1:25

score 494 · Answer 12 · 2026-04-24 03:47:42

494

Views

$\lim (x_n^k) = (\lim x_n)^k$

Published on 24 Apr 2026 - 3:47

score 60 · Answer 13 · 2026-04-13 14:04:56

60

Views

how to show that $\sum_{ n=1}^{∞ }a_n^+ = ∞ $?

Published on 13 Apr 2026 - 14:04

score 124 · Answer 14 · 2026-05-10 14:50:52

124

Views

Is this condition equivalent to being harmonic?

Published on 10 May 2026 - 14:50

score 399 · Answer 15 · 2026-05-15 08:38:20

399

Views

Doubt in proof of Convex Function Continuous

Published on 15 May 2026 - 8:38

List Question

If $f_n\to f$ in $L^p$ and $g_n$ is uniformly bounded and $g_n\to g$ a.e. then $g_nf_n\to gf$ in $L^p$.

My Proof: Every convergent sequence is a Cauchy sequence.

Let $f:[a,b]\to \mathbb{R}$ is integrable on $[a,b]$. Show that $f_1:[a,b]\to \mathbb{R}$ such that $f_1(x)=\sup\{f(x), 0\}$, is integrable.

How to prove this specific function has limit either $\infty$ or 0?

Whether the indefinite integral of a differentiable function is differentiable or not

Chacking integrability of a piecewise function

Prove or disprove that $(b-a)^2(\log(b)-\log(a))^{-2}\ge ab$ for all $a,b>0$

Finding a function that solves the condition

proof $\lim x_n \le \lim y_n$ if $x_n \le y_n$ (clarification)

Monotone convergence theorem implies Nested Interval property

Exponentials inequality: $|e^x + e^y| \leq |x - y|$ for $x,y<0$.

$\lim (x_n^k) = (\lim x_n)^k$

how to show that $\sum_{ n=1}^{∞ }a_n^+ = ∞ $?

Is this condition equivalent to being harmonic?

Doubt in proof of Convex Function Continuous

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