A convergent sequence is a sequence having finite limit. For example, {1/n} is a convergent sequence with limit 0. Here, we study the definition and examples of convergent sequence. Convergent Sequence Definition A sequence {xn} is said be convergent if it has a finite limit L. That is, for every ε>0, there is a natural … Read more
An unbounded sequence is a sequence which is not bounded. For example the sequence {n} of natural numbers is unbounded. In this article, we study unbounded sequences with its definition and examples. Unbounded Sequence Definition A sequence {an} is said to be unbounded if there is no real number M such that |an| ≤ M. … Read more
No, a continuous function is not always differentiable. For example, f(x)=|x| is continuous but not differentiable at x=0. In this post, we will study the following: is a continuous function always differentiable? A continuous function may not be differentiable always. Although, a differentiable function is always continuous. Lets prove this fact here. Recall the definitions … Read more
In this article, we prove that a convergent sequence is bounded. A sequence is said to be convergent if it has finite limit. Question: Prove that A Convergent Sequence is Bounded. Proof Let us assume that {xn} is a convergent sequence. So it has finite limit, say L. That is, limn→∞ xn = L. So … Read more
A sequence {xn} is called a bounded sequence if k ≤ xn ≤ K for all natural numbers n. For example, {1/n} is a bounded sequence since 0 < 1/n ≤ 1 for all n. Let us now learn about bounded sequence. Bounded Sequence Definition A sequence {xn} is said to be bounded if k … Read more