Question

Question about the Mathematical Real Analysis Proof

Show that if xn → 0 then √xn → 0.

Proof. Let ε > 0 be arbitrary. Since xn → 0 there is some N ∈N such that |xn| < ε^2 for all n > N. Then for all n > N we have that |√xn| < ε

**My question is based on the sequence convergence
definition it should be absolute an-a<ε but here
why we can take xn<ε^2 rather than ε?**

**Please explain and draw the number line to explain
ε.**

Answer #1

Mathematical Real Analysis
Convergence Sequence Conception Question:
By the definition: for all epsilon >0 there exists a N such
that for all n>N absolute an-a < epsilon
Question: assume bn ----->b and b is not 0. prove
that lim n to infinity 1/ bn = 1/ b .
Please Tell me why here we need to have N1 and N2 and
Find the Max(N1,N2) but other example we don't.
Solve the question step by step as well

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In this task, you will write a proof to analyze the limit of a
sequence.
ASSUMPTIONS
Definition: A sequence {an} for n = 1 to ∞ converges
to a real number A if and only if for each ε > 0 there is a
positive integer N such that for all n ≥
N, |an – A| < ε .
Let P be 6. and Let Q be 24.
Define your sequence to be an = 4 +
1/(Pn +...

In this task, you will write a proof to analyze the limit of a
sequence.
ASSUMPTIONS
Definition: A sequence {an} for n = 1 to ∞ converges
to a real number A if and only if for each ε > 0 there is a
positive integer N such that for all n ≥
N, |an – A| < ε .
Let P be 6. and Let Q be 24.
Define your sequence to be an = 4 +
1/(Pn +...

Claim: If (sn) is any sequence of real numbers with
??+1 = ??2 + 3?? for
all n in N, then ?? ≥ 0 for all n in N.
Proof: Suppose (sn) is any sequence of real numbers
with ??+1 = ??2 + 3??
for all n in N. Let P(n) be the inequality statements ??
≥ 0.
Let k be in N and suppose P(k) is true: Suppose ?? ≥
0.
Note that ??+1 = ??2 +
3?? =...

Mathematical Real Analysis Questions
You have to answer two questions in order to get a
thumb's up and good
Q.1 Let A = (0,2]. Prove that A does not have a
minimum. What is the inﬁmum of A?
Q.2. Theorem. Given any two real numbers x < y, there exists
an irrational number satisfying x <t< y.
Proof. It follows from x < y that x−√2 < y−√2. Since Q is
dense in R, there exists p ∈Q such that x−√2...

Do not use binomial theorem for this!! (Real analysis
question)
a) Let (sn) be the sequence deﬁned by sn = (1 +1/n)^(n). Prove
that sn is an increasing sequence with sn < 3 for all n.
Conclude that (sn) is convergent. The limit of (sn) is referred to
as e and is used as the base for natural logarithms.
b)Use the result above to ﬁnd the limit of the sequences: sn =
(1 +1/n)^(2n)
c)sn = (1+1/n)^(n-1)

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