My questions are: •What is Xn? ・Why lim n→∞{f(Xn)} ≦lim n→∞{g(Xn)}?Any Thm about sequences? imil ov 2.5 THEOREM Suppose f :D R and g : D R, xo is an accumulation point of D, and f and g have limits at...

My questions are: •What is Xn? ・Why lim n→∞{f(Xn)} ≦lim n→∞{g(Xn)}?Any Thm about sequences?

Extracted text: imil ov 2.5 THEOREM Suppose f :D R and g : D R, xo is an accumulation point of D, and f and g have limits at xo. If f(x) < g(x)="" for="" all="" x="" e="" d,="" then="" lim="" f(x)="">< lim="" g(x).="" xー→x0="">


Extracted text: 20. Prove Theorem 2.5: Suppose f : D → R and g : D → R, xo is an accumulation point of D, and f and g have limits at xo. If f(x)< g(x)="" for="" all="" x="" €="" d,="" then="" lim="" f(x)="">< lim="" g(x).="" then="" {f(xn)}1="" let="" {xn}="1" be="" a="" sequence="" of="" points="" in="" d="" coverging="" to="" xo="" .="" converges="" to="" lim,-zo="" f(x),="" and="" {g(xn)}1="" converges="" to="" lim,-zo="" 9(x).="" applying="" a="" previous="" result="" about="" sequences="" gives="" lim="" f(x)="lim" {f(xn)}="">< lim="" {g(x,n)}="lim" g(x).="">