Recent content by ricardianequiva

yeah... Thanks for the help though

Hmm we haven't done differentiation yet so I'm not sure how helpful the |(f(x)-f(a))/(x-a)| will be.

Homework Statement Show, from the definition of continuity, that the power series function f(x)=sum(a_n*x^n) is continuous for its radius of convergence.Homework Equations Definition of continuityThe Attempt at a Solution Must show that for any |a| < R, given e>0 there exists d>0 such that...

Homework Statement Let f:R->R be a continuous function where limit as x goes to positive/negative infinity is negative infinity. Prove that f has a maximum value on R. Homework Equations None The Attempt at a Solution I tried to use the definition of infinite limits but I'm not...

Homework Statement Prove that the following sequence is Cauchy: a_n = [a_(n-1) + a_(n-2)]/2 (i.e. the average of the last two), where a_0 = x a_1 = y Homework Equations None The Attempt at a Solution I was trying to use the definition of Cauchy (i.e. |a_m - a_n| < e) by...

in general, how does one go about proving limits of recursive sequences using the definition? for example, how does one prove a_n+1 = (a_n)^2/5 => lim(a_n)=0? For me it's obvious but the TA insists that things like that require proving?

yes, i can see how that shows that the limit is 0, but i don't think its rigorous enough. is there some way to actually show that the limit is 0, perhaps using definitions of limits?

yes. but i don't see how that helps

Homework Statement Let the sequence {a_n} defined by: a_n+1 = a_n/[sqrt(0.5a_n + 1) + 1] Prove that {a_n} converges to 0 Homework Equations The Attempt at a Solution I tried manipulating the equation but to no avail...

One more question: In our textbook we are given a theorem that: If f o g is bijective then g is injective and f is surjective. I can informally see this by drawing Venn Diagrams, but how would one go about doing a formal proof.

got it, thanks!

O I made a mistake in the question, we are given that h o g o f is bijective.

No there aren't any other conditions.

Homework Statement Consider arbitrary sets A, B, C and D with arbitrary functions: f:A-->B, g:B-->C, h:C-->D. We define a composite function h o g o f:A-->D. Given that h, f, and h o g o f are bijective, and g is injective, show that g is also surjective (i.e. g is bijective). This...