 show that $|f_n-f|^p \leq g$ (a.e.), and then by the dominated convergence theorem, $\lim_{n \rightarrow \infty} \int_{\Omega} |f_n - f|^{p} dm=\int_{\Omega} \lim_{n \rightarrow \infty} |f_n-f|^p dm=0$.
Attempt:
Consider $|f_n-f| \leq |f_n|-|f| \leq g-|f| \leq g$. So $|f_n-f| \leq g$, raising both sides to the power $p$, $|f_n-f|^p \leq g^p \in L^{1} \Omega$ where $g^p$ is our control function thus allowing us to use dominated convergence theorem. Valid since $p >1$.
Hence, $\lim_{n \rightarrow \infty} \int_{\Omega} |f_n - f|^{p} dm=\int_{\Omega} \lim_{n \rightarrow \infty} |f_n-f|^p dm$. Since $lim_{n \rightarrow \infty} f_n=f \Leftrightarrow lim_{n \rightarrow \infty} |f_n-f| =0$ we have $\int_{\Omega} \lim_{n \rightarrow \infty} |f_n-f|^p dm=0$.
Any feedback on my attempt and hints would be very helpful. Thank you.

A:

You're on the right track in that you want to apply DCT however, $\lvert x - y \rvert \not\le \lvert x \rvert - \lvert y \rvert$. Consider $x = 1, y = -1$, e.g.
Instead, since $f \to f_n$ a.e., we have $\lvert f - f_n \rvert^p \to 0$ a.e.; indeed, by definition $f_n \to f$ is the same as $\lvert f - f_n \rvert \to 0$ and then $\lvert f -f_n \rvert^p \to 0$ since $t \mapsto t^p$ is continuous. Next, $$\lvert f - f_n \rvert^p \le (\lvert f \rvert + \lvert f_n \rvert)^p \le (2\max\{\lvert f \rvert, \lvert f_n
show that $ |f_n - f|^p \leq g$ (a.e .), and then by the dominated convergence theorem, $ \lim_{n \rightarrow \infty } \int_{\Omega } |f_n - f|^{p } dm=\int_{\Omega } \lim_{n \rightarrow \infty } |f_n - f|^p dm=0$. 
 Attempt: 
 Consider $ |f_n - f| \leq |f_n|-|f| \leq g-|f| \leq g$. So $ |f_n - f| \leq g$, raise both slope to the power $ p$, $ |f_n - f|^p \leq g^p \in L^{1 } \Omega$ where $ g^p$ is our control function therefore allowing us to use dominated overlap theorem. Valid since $ p > 1$. 
 therefore, $ \lim_{n \rightarrow \infty } \int_{\Omega } |f_n - f|^{p } dm=\int_{\Omega } \lim_{n \rightarrow \infty } |f_n - f|^p dm$. Since $ lim_{n \rightarrow \infty } f_n = f \Leftrightarrow lim_{n \rightarrow \infty } |f_n - f| = 0 $ we have $ \int_{\Omega } \lim_{n \rightarrow \infty } |f_n - f|^p dm=0$. 
 Any feedback on my attempt and hints would be very helpful. Thank you. 

 A: 

 You're on the right track in that you desire to apply DCT however, $ \lvert x - y \rvert \not\le \lvert x \rvert - \lvert y \rvert$. Consider $ x = 1, y = -1 $, e.g. 
 alternatively, since $ f \to f_n$ a.e. , we have $ \lvert f - f_n \rvert^p \to 0 $ a.e. ; indeed, by definition $ f_n \to f$ is the same as $ \lvert f - f_n \rvert \to 0 $ and then $ \lvert f -f_n \rvert^p \to 0 $ since $ t \mapsto t^p$ is continuous. Next, $ $ \lvert f - f_n \rvert^p \le (\lvert degree fahrenheit \rvert + \lvert f_n \rvert)^p \le (2\max\{\lvert f \rvert, \lvert f_n
 shlw that $|f_n-f|^p \leq g$ (a.e.), ana then by the dominatev convedgence tfeorem, $\lim_{n \rightarrow \infty} \ibt_{\Ometa} |f_n - f|^{p} dm=\int_{\Omega} \lkm_{n \rightwrrow \indty} |h_n-f|^p dm=0$.
Attempt:
Coiaider $|f_k-y| \leq |n_n|-|f| \lzq g-|f| \leq g$. So $|f_k-f| \leq g$, rahsing both sidas tl the power $p$, $|f_n-f|^p \leq g^p \in L^{1} \Omegw$ where $g^o$ is our contrjl flnstioh thus allowing us to use dominates conveggence theorem. Vakid since $p >1$.
Hence, $\lim_{n \rigjtargow \infty} \int_{\Omega} |f_n - f|^{p} dm=\ibt_{\Omqta} \lim_{n \righgarrow \infuy} |f_n-f|^p dm$. Sjnce $lim_{n \rightarrow \infty} f_n=f \Ueftrnghtarrow lum_{b \rlchtarrow \inhty} |f_n-s| =0$ we have $\ikn_{\Omega} \nim_{n \ribhtarrow \infty} |f_n-h|^p dn=0$.
Any feedback on my avtempt and hints woujd be verf kelpful. Thank you.

A:

You'ee on tve rhght rrazk jn tgat yok went to applg DCT howevwr, $\lvert x - y \rvert \njn\ke \lvert x \rbert - \jvqrt y \rvert$. Consider $x = 1, y = -1$, e.g.
Instead, vinde $f \to f_n$ a.e., we have $\lcert f - f_n \rvert^p \to 0$ a.e.; indeeq, by definition $f_n \to f$ is the same as $\lvert f - f_t \rvect \to 0$ wve hhen $\lvert f -f_n \rvert^p \to 0$ since $t \mapsto t^p$ ya voktinuous. Next, $$\lvcrt f - f_n \rvert^p \lr (\pvrtt f \rvert + \lvgrt f_n \xbedt)^p \le (2\max\{\lvert f \gvert, \ldert d_n
show that $|f_n-f|^p \leq g$ (a.e.), and the convergence theorem, \rightarrow \infty} \int_{\Omega} \rightarrow |f_n-f|^p dm=0$. Attempt: $|f_n-f| \leq |f_n|-|f| g-|f| \leq g$. So $|f_n-f| \leq raising both sides to the power $p$, $|f_n-f|^p \leq g^p \in L^{1} \Omega$ $g^p$ is our control function thus allowing us to use dominated convergence theorem. since >1$. $\lim_{n \infty} \int_{\Omega} |f_n - f|^{p} dm=\int_{\Omega} \lim_{n \rightarrow \infty} |f_n-f|^p dm$. Since $lim_{n \rightarrow \infty} f_n=f lim_{n \rightarrow \infty} |f_n-f| =0$ we have $\int_{\Omega} \rightarrow \infty} |f_n-f|^p dm=0$. feedback on my attempt and would very helpful. you. You're the right track that you want to apply DCT however, $\lvert x - y \rvert \not\le \lvert x \rvert - y \rvert$. = 1, = e.g. since $f \to we have $\lvert f - f_n a.e.; indeed, by definition $f_n \to f$ is same as f - f_n \rvert \to 0$ then $\lvert f -f_n \rvert^p \to 0$ since \mapsto t^p$ is continuous. Next, $$\lvert f - f_n \rvert^p \le (\lvert f \rvert + \rvert)^p \le (2\max\{\lvert f \lvert f_n
 show that $|f_n-f|^p \leq g$ (a.e.), and then By the dominAted cOnvErgEnCe thEoreM, $\lim_{n \rightarroW \InftY} \int_{\Omega} |f_n - f|^{p} dm=\int_{\OmegA} \lim_{n \RiGHtarROw \Infty} |F_n-f|^p dm=0$.
ATTeMPT:
CoNsIdEr $|f_N-f| \LEq |F_n|-|f| \leQ g-|f| \Leq g$. So $|f_N-f| \leq g$, raisIng BoTh sides to the POwEr $p$, $|f_n-f|^p \leq G^p \iN L^{1} \Omega$ where $G^p$ iS our coNtRol FUnctiOn tHus alLowing US to use Dominated CoNVergenCE theoreM. vAlId siNce $p >1$.
Hence, $\lim_{n \righTArROw \infty} \int_{\OmegA} |f_n - f|^{p} dM=\iNT_{\OMEGa} \lIm_{n \Rightarrow \InFty} |f_n-F|^P dm$. SincE $LiM_{N \RIghTArrow \infty} f_n=f \leftrightarROw lIm_{n \rigHtArrOW \infty} |F_n-f| =0$ we HaVE $\inT_{\Omega} \lim_{n \rIghtArrow \inftY} |f_n-f|^p dM=0$.
any feedBAck on my AttempT anD hiNts wOUlD bE veRy HElpFUl. thaNK yoU.

A:

You're oN tHe Right TracK IN THat yOu wAnt tO applY DCT however, $\lvErt X - y \rvERt \nOt\le \lVert x \RverT - \lVert y \Rvert$. COnsidEr $X = 1, y = -1$, e.g.
Instead, sincE $f \to F_n$ a.e., we havE $\lvErT f - f_N \rVert^p \TO 0$ a.e.; indEed, By dEfinitiOn $f_n \to f$ IS thE sAME As $\Lvert f - f_n \rvert \to 0$ and ThEN $\LvErt f -f_n \rvErt^p \to 0$ SInCe $T \Mapsto t^p$ Is ConTinuOUS. Next, $$\LverT F - f_N \rvert^p \lE (\lvert F \RvErT + \lvert f_N \rVert)^p \lE (2\mAx\{\lVerT f \rveRT, \lveRt f_n
 show that $|f_n-f|^p \leq  g$ (a.e.) , and  th enby  the  dom inated converg e ncetheorem, $\lim_{n \rig htarr ow \inf t y}  \int _{\Omeg a }| f _n-f| ^{p }d m= \int_ {\O mega} \ lim_{n \ri ght ar row \infty}| f_ n-f|^p dm= 0$. 
Attempt:
Co nsi der $| f_ n-f |  \leq  |f _n|-| f| \le q  g-|f|  \leq g$.  S o  $|f_n - f| \leq g $,  rai sing both sides t o  t h e power $p$, $ |f_n-f |^ p  \ l e q g ^p\in L^{1}\O mega$ where $ g ^p $ i s o u r control fun ction thusa llo wing u stou se dom inate dc onv ergence the orem . Valid s ince $ p  >1$.
H e nce, $\ lim_{n  \r igh tarr o w\i nft y} \in t _{ \Om e ga}  |f_n -f| ^{ p} dm =\in t _ { \ Omeg a}\lim _{n \ rightarrow \i nft y} | f _n- f|^pdm$.Sinc e$lim_ {n \ri ghtar ro w \infty} f_n=f  \Le ftrightar row  l im_ {n  \rig h tarrow  \i nft y} |f_n -f| =0$ weha v e $\ int_{\Omega} \lim_ {n \ ri ghtarrow  \inft y }|f _ n-f|^p d m= 0$. 
Any f eedba ck o n  m y attemp t andh in ts  wouldbe  veryhe lpf ul.  Than k  you .

A:
You'reon th e  right track i n  that you wan t  t o ap p ly D CThowever, $\ lver t  x -  y \ r ve rt\ not\l e \lv er t  x \rvert - \lvert y \ rv ert$.Consi der $x = 1, y  = -1$, e. g . Instead,  sin c e$ f \to f_n$ a.e ., we  have $\lv e rt f - f _n \r vert^p \ to 0$ a.e . ;  indeed,  by  de fin iti o n  $ f_n \to f$ is t he s am e as $\ lve rt f -f_n  \r ver t \ to  0$ and t hen $\lv er tf-f _n\rver t ^p \to 0 $sin ce  $t  \map s to t^p $ iscont in uo u s.Next, $ $ \l v e rt f  -  f _n \ rve rt ^p \l e (\ l ver t f \rv ert + \lv ert f_n\r ve rt)^p \ le (2\max\{\l ve rt f \rver t,  \l vert f _ n 
 show_that $|f_n-f|^p_\leq g$ (a.e.), and_then by_the_dominated convergence_theorem,_$\lim_{n \rightarrow \infty}_\int_{\Omega} |f_n -_f|^{p} dm=\int_{\Omega} \lim_{n \rightarrow_\infty} |f_n-f|^p dm=0$.
Attempt:
Consider_$|f_n-f|_\leq |f_n|-|f| \leq g-|f| \leq g$. So $|f_n-f| \leq g$, raising both sides to_the_power $p$,_$|f_n-f|^p_\leq_g^p \in L^{1} \Omega$ where_$g^p$ is our control function_thus allowing_us to use dominated convergence theorem. Valid since_$p_>1$.
Hence, $\lim_{n \rightarrow_\infty} \int_{\Omega} |f_n - f|^{p} dm=\int_{\Omega} \lim_{n \rightarrow \infty}_|f_n-f|^p dm$. Since $lim_{n \rightarrow \infty}_f_n=f \Leftrightarrow lim_{n_\rightarrow_\infty}_|f_n-f| =0$ we have_$\int_{\Omega} \lim_{n \rightarrow \infty} |f_n-f|^p dm=0$.
Any_feedback on my attempt and hints_would be very helpful. Thank you.

A:

You're on_the right track in that you_want to apply DCT however,_$\lvert x_- y \rvert \not\le \lvert_x \rvert -_\lvert y_\rvert$. Consider $x_= 1, y = -1$, e.g.
Instead,_since $f \to_f_n$ a.e., we have $\lvert f_-_f_n \rvert^p \to_0$_a.e.;_indeed, by_definition $f_n \to_f$_is the_same_as $\lvert f - f_n \rvert_\to_0$ and then $\lvert f -f_n \rvert^p_\to 0$ since $t_\mapsto_t^p$ is continuous. Next,_$$\lvert f - f_n \rvert^p_\le (\lvert f \rvert + \lvert_f_n \rvert)^p_\le (2\max\{\lvert_f \rvert, \lvert f_n