Question

Let f:D + R with , and accumulation point of D. f has a limit at zo if and only if for each sequence {In}n=1 converging to to with In E D and In + x, for all n, the sequence {f(In)}=1 converges. Theorem 2.1, P. 69

Let f : D → IR with x0 and accumulation point of D. f has a limit at x0 if and only if for each sequence {xn} ∞ n=1 converging to x0 with xn ∈ D and xn 6= x0 for all n, the sequence {f(xn)} ∞ n=1 converges.

Answer 1

f has a limit at to * lim gen) =LER. ako XED > For any & >o, Is>osit whenever If ()-L) <E. drx, xo)<S for someren It {xu3 is any seg. in D convenging to to, with 27to theN IN EN Sot dfang 2.) < 5 t n >N Then If (an) -L/ <e an>N from)}" converses to h in R. horo .. For any sequence in D converging to the the image of that sequence converges in R.. conversely, for any seg, sono in D with 4m x = % f xa txa MEN, {from to converges. Let &xo be be given. Since fix a seg. 3xudu (as above) ina. n=1 ent

Since fexus} converses in R L. texn) ve E30, »hive {>o, & D is complete, FLER st 7 NEIN st 12 (xx) tk E whenever n>N. I NEIN s.t if(w)rtka given . The het su be since lim xa = Ro I NEIN sit d (Rag Ro) < SENIN let x be any pt in D with x 7o, Ix-xocs. since to is an accumulation pt. of D, we can choose a sequena syn} wa in D, oy verging to xo , with - Inox for some no»N2. Since both ren} & {{n} convenge to to, arxasty) 0. (. dcxm, yn) s dcem, xs) +%, yu) od Iteyn) -61 S1fCY93-12mXn-L! < If )-f(xw) +42 EndNg simet is continuous and d cau, yn) so, Iseen)-HX)/ ..7 N3 sit lfeynd-resens / <£,&n>N3. ... lfrund-L1 <cit & e , n >N where N = man { NI, N2, N3} &ncn. flynd-11ce (": No >N) - If (x2-L1 LE => whenever Ix-xol <s, with x GD 1, 203, fiat LIKE has a limit at ko.