Recent content by ILikePizza

Homework Statement Given that f(x + y) = f(x) + f(y), prove that (a) if this function is continuous at some point p, then it is continuous everywhere (b) this function is linear if f(1) is continuous. Homework Equations definition of continuity The Attempt at a Solution (a) I...

Homework Statement Suppose that (x_n) is a properly divergent sequence, and suppose that (x_n) is unbounded above. Suppose that there exists a sequence (y_n) such that limit (x_n * y_n) exists. Prove that (y_n) ===> 0. Homework Equations (x_n) ===> 0 <====> (1/x_n) ===> 0...

So x-c = \delta, and x2+ cx+ c2 --> 3c^2. So if \delta = \frac{\epsilon}{3c^2}, that's good. But how do I write this rigorously? Thanks again

Homework Statement Prove that as x --> c, lim (x^2) = (c^2) using only the definition. What does this tell you about x --> c, lim (x^3) = (c^3)? x --> c, lim (x^4) = (c^4) ? Prove it. Homework Equations The definition of a limit. The Attempt at a Solution Let $\epsilon > 0$ be...

L+1/L = x_n = L x _n+1 > L, a contradiction? Is that what you were fishing for?

Thanks for the reply. I am aware that this will work, but I still have no idea where to start. It is easy to show that this sequence is monotone increasing, but what about the second part? That's where I'm stuck. Thanks

Homework Statement (x_n) is a sequence and x_1 > 2. From then on, x_{n+1} = x_n + 1/x_n Prove that (x_n) is divergent. Homework Equations n/a The Attempt at a Solution I first tried assuming that a limit existed, but I didn't get a contradiction. (I had x = 2 + 1/x, x = (2 \pm...

I don't think you guys are reading what i said... I now *know* that .9999 = 1, but my question is: "I guess that my real question is that if a1.a2a3a4...aN = d1.d2d3d4...bN (finite number of digits) then all the corresponding digits are the same." Is this true or not? Do you have a proof?

So, Friday in class, my teacher was talking about how .99999... = 1.00000... So, this makes sense and all, but this is my question: How do you know that 1.230000... = 1.230000...? I guess that my real question is that if a1.a2a3a4...aN = d1.d2d3d4...bN (finite number of digits) then...