This is what you previously said.
Did you perhaps mean it is bounded in [a[SIZE="3"]+1/n,b[SIZE="3"]-1/n]? The signs on the 1/n terms are different between your first post and your more recent one.
I did not use that logic in my proof, however. Aside from that, then, does my work make sense?
I'm not familiar with this assertion: if f is continuous, it is bounded in [a-1/n,b+1/n]
For instance, f = 1/x is continuous in (0,1) but is unbounded and discontinuous in the closed interval [0,1].
But, let me try to follow other than that...
For the lower endpoint:
Since f is...
Homework Statement
Suppose that the function f|(a,b)→ℝ is uniformly continuous. Prove that f|(a,b)→ℝ is bounded.
Homework Equations
A function f|D→ℝ is uniformly continuous provided that whenever {un} and {vn} are sequences in D such that lim (n→∞) [un-vn] = 0, then lim (n→∞) [f(un) -...
I never claimed it was a theorem, and indeed referenced it as part of an example. But, yes, I can see where it was misleading.
My question stands, though. How is that sufficient justification, even within the example I've scanned? It seems to be glossing over important steps such as...
Sorry, I copied the problem incorrectly. The piecewise function should read:
f(x) = x^2, if x<= 0
x+1, if x>0
Thus, again:
Check x=0. {1/n} converges to 0. {f(1/n)} converges to 1. f(0) = 0. 0 ≠ 1. Thus the function is not continuous at x=0.
This would be correct if I was trying to prove...
Homework Statement
Define
f = { x^2 if x \geq 0
x if x < 0
At what points is the function f | \Re -> \Re continous? Justify your answer.
Homework Equations
A function f from D to R is continuous at x0 in D provided that whenever {xn} is a sequence in D that converges to x0, the...
Aghhhh, so simple! Something about the way I do proofs is just <i>wrong</i>, I always get caught in roadblocks of thinking and miss simple detours like that.
s_n \leq M*r + M*r^2 + ... + M*r^n
Then s_n \leq M(r-r^{n+1})/(1-r)
Then s_n \leq Mr/(1-r) = M'
Then |s_n| \leq M' for all natural...
Homework Statement
Let b_n be a bounded sequence of nonnegative numbers. Let r be a number such that 0 \leq r < 1.
Define s_n = b_1*r + b_2*r^2 + ... + b_n*r^n, for all natural numbers n.
Prove that {s_n} converges.
Homework Equations
Sum of first n terms of geometric series = sum_n...