4. Write the negation of each statement.(a) The function f(x) =x -9 is continuous at x = 3.(b) The relation R is reflexive or symmetric.(c) Four and nine are relatively prime.(d) x is in A or x is not in B.(e) Ifx<7, then f(x) is nọt in C.(f) If(a,) is convergent, then (a,) is monotone and bounded.then f(A) is open.(g) If f is continuous and A is open, then f'(A) is open.

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Answered: (f) If (a„) is convergent, then (a,) is…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let (fa)n be a sequence of functions fa : [0, 1] Rdefined by fa(z)= 2nz +1 , if r< , if SIS1 Then 1. (f.) doesn't converges to 0 pointwisely on 0, 1]. 2 f) coverges to f =0 uniformly on [0, 1] and thus fa(z)dr converges to f(z)dr. 3. (f.) converges to 0 pointwisely on [0, 1] but not uniformly. 4. (f.) converges to f =0 uniformly on 0, 1] and so (S.) converges to 0 uniformly. The segur nce of functions on 01 Then on 0. 1 we have 75°F Farch TOSHIBA 近 4.
Suppose that 7 is a Riemann integrable function on [", $]. Prove, in two lines only, that if 7 is continuous on [", $] then 7 is bounded on [", $].
dssignment: 1. Show that the sequence Pfo}, where fex) = tan hx, xzo in uniprmly convergent on ony Interral [a,h], aso but is only pointwise conrergent on [0,6]. Hint: (tonhx + cutnix = ). %3D ノ 2. Show that the sequence fh}, where farx) = is uniformly convergent nx+1 on [a,L], a >o but in only pointwise Converg ent on [0,6]. If fcx) = Sin nz OKx8I, does there exist KEN Such that Ans. K =1l
8. Prove that lim(x,) =0 if and only if lim()x,1) = 0. Give an example to show that the conver- gence of (1x, 1) need not imply the convergence of (x,).
$$\begin{array}{l}\text { For every } n \in \mathbb{N}^{*} \text { we consider the } \\\text { function } f_{n}:(1 ; \infty) \rightarrow \mathbb{R} \\\quad f_{n}(x)=\frac{\ln ^{n} x}{x} \\\text { sMow that } f_{n}(x) \leqslant \frac{n^{n}}{e^{n}} \forall x \in(1, \infty) \\\text { and } \forall n \in M^{*}For every $n \in \mathbb{N}^{*}$ we consider thefunction $f_{n}:(1 ; \infty) \rightarrow \mathbb{R}$$\quad f_{n}(x)=\frac{\ln ^{n} x}{x}$sMow that $f_{n}(x) \leqslant \frac{n^{n}}{e^{n}} \forall x \in(1, \infty)$and $\forall n \in M^{*}$\end{array}$$ please check the attached image to see the problem clearly,I have attached a screenshot for the output of the above latex code.
29) Identifa of convergen'ce af a power seri's Ž (-3) nx the itervel n-l Ansus: a ) b) c) d) -32x23 e) 3
Let f be continuous on [-1,5], where f(-1) = 12 and f(5) = 6. Can we conclude that a value -1 < c < 5 exists such that f(c) = 2?
3. Let {gn} be a sequence of integrable functions such that gn →g a.e. with g integrable. Show that lim f [gn – g| = 0 if and only if lim f \gn| = S \g].
Let (1m) = (0,5,0.5.0) be sequence in R, dis discrete metric then (xn) is neither Cauchy nor converges Cauchy and converges Cauchy but not converges not Cauchy but converges
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(f) If (a„) is convergent, then (a,) is monotone and bounded. (g) If f is continuous and A is open, then f'(A) is open.