## How do you prove a sequence is not Cauchy?

For a sequence not to be Cauchy, there needs to be some N > 0 N>0 N>0 such that for any ϵ > 0 \epsilon>0 ϵ>0, there are m , n > N m,n>N m,n>N with ∣ a n − a m ∣ > ϵ |a_n-a_m|>\epsilon ∣an−am∣>ϵ.

**Why is n not a Cauchy sequence?**

Consider an = (−1)n and take ϵ = 1/2 and set m = n + 1. Then for all N, if n, m ≥ N we have |an − am| = |an − an+1| = |2| ≥ 1/2 = ϵ, so the sequence is not Cauchy.

### How do you prove a function is Cauchy?

The proof is essentially the same as the corresponding result for convergent sequences. Any convergent sequence is a Cauchy sequence. If (an)→ α then given ε > 0 choose N so that if n > N we have |an- α| < ε. Then if m, n > N we have |am- an| = |(am- α) – (am- α)| ≤ |am- α| + |am- α| < 2ε.

**How do you prove Cauchy criterion?**

If a sequence (xn) converges then it satisfies the Cauchy’s criterion: for ϵ > 0, there exists N such that |xn − xm| < ϵ for all n, m ≥ N. If a sequence converges then the elements of the sequence get close to the limit as n increases.

## Is Cauchy a 1 N sequence?

Thus, xn = 1 n is a Cauchy sequence.

**Why is sqrt n not Cauchy?**

As the elements of {√n} get further apart from each other as n increase this is clearly not Cauchy. Finding a counter example is simply a matter of finding m>n so that √m−√n>ϵ. Or in other words √m>√n+ϵ. Or in other words m>(√n+ϵ)2=n+2√nϵ+ϵ2….

### Is Root N Cauchy?

As the elements of {√n} get further apart from each other as n increase this is clearly not Cauchy.

**Are all convergent sequences Cauchy?**

Every convergent sequence is a Cauchy sequence. The converse statement is not true in general. However, in the metric space of complex or real numbers the converse is true.

## Are all Cauchy sequences convergent?

Every real Cauchy sequence is convergent. Theorem. Every complex Cauchy sequence is convergent.

**What is Cauchy general principle?**

Cauchy’s general principle of convergence: An infinite series x n converges iff for every ε > 0, there exists a positive integer N such that │ xn1 ……. xm │< ε whenever m ≥ n ≥ N.

### Is series 1 N convergent?

What is the summation of 1/n converge to? 1/n is a harmonic series and it is well known that though the nth Term goes to zero as n tends to infinity, the summation of this series doesn’t converge but it goes to infinity. It’s not very difficult to prove it.

**Is the square root of a Cauchy sequence Cauchy?**

Sequence of Square Roots of Natural Numbers is not Cauchy.