use the definition to prove that
We have:
Now, multiplying both sides by and relabelling the
index gives us:
use the definition to prove that By definition of the Bernoulli numbers: no xcscx 2 (1)...
1. Use the definition of limits to prove that
5. (a) (7 points) Use the definition of convergence to prove that the sequence {(-1)-+ 히 converges to 0 (b) (7 points) Prove that the sequence k=1 does not converge.
(3) Use the definition of convergence to prove each of the following (a) 1 is not the limit of the sequence sn (-1)" (b) lim = 1/2 2n (c) Suppose that lim an = a. Prove that lim 3 . an За.
Determine a definition for what it means to add two real numbers. Then, prove that the real numbers are closed under addition.
Exercise 3. (i) Use the definition (of a maximum element of a set) to prove that if a set of real numbers has a maximum element then this element is unique. (ii) Prove that a finite non-empty set of real numbers has a minimum element (Remark: This is part of a Proposition from class.)
The Lucas numbers L(n) have almost the same definition as the Fibonacci numbers: If n = 1 if n- 2 L(n 1) L(n - 2) if n > 2. 12, as in Theorem 3.6. Prove that L(n)-α, β n for all n E N. Use strong induction Let α = 1 + v/5 and β-- Proof. First, note that and L(2) suppose as inductive hypothesis that L()-α4 β, for all i k, for some k > 2. Then l(k) =...
1. Use only definition 4.2.1 (functional limit) to prove that . lim (2.rº + 3.2 + 4) = 6
how do I prove this by assuming true for K and then proving
for k+1
Use mathematical induction to prove that 2"-1< n! for all natural numbers n.
Use mathematical induction to prove that 2"-1
Use the definition to prove lim x +4 - X -1 2x + 1
2. Use the ε - δ definition for the limit to prove that limx→-2 (4x - 3) = -113. Use the limit definition of the derivative to find the derivative of the function f(x) = √(4x + 1)4. Find the equation of the tangent line to the curve ve y = (1 + 2x) 10 at the point (-1,1).