Knowee
Questions
Features
Study Tools

Let (xn)n≥1 and (yn)n≥1 be bounded sequence. Prove thatlim infn→∞ (xn + yn) ≥ lim infn→∞ xn + lim infn→∞ yn

Question

Let (xn)n1 (x_n)_{n \geq 1} and (yn)n1 (y_n)_{n \geq 1} be bounded sequence. Prove that ( \liminf_{n \to \infty} (x_n + y_n) \geq \liminf_{n \to \infty} x_n + \liminf_{n \to \infty} y_n. \

🧐 Not the exact question you are looking for?Go ask a question

Solution

Sure, here is the proof:

Step 1: Definitions First, let's define what we mean by lim inf. The limit inferior (or infimum limit) of a sequence (xn) is the greatest lower bound of the set of all limit points of the sequence. In other words, it's the largest value that we can guarantee the sequence wi

*********************************************** ************************************************ ************************************************ ************************************************ ************************************************ ************************************************ ************************************************ ************************************************ *********

This problem has been solved

Similar Questions

Let (xn)n≥1 and (yn)n≥1 be bounded sequence. Prove thatlim infn→∞ (xn + yn) ≥ lim infn→∞ xn + lim infn→∞ yn

Let (xn) be a bounded sequence in R. Show that there exist subsequences (xnk ) and(xmk ) of (xn) such thatlimk→∞ xnk = lim sup xn and limk→∞ xmk = lim inf xn.

Define (xn)n≥1 by x1 := 0 and xn+1 := (xn + 3)/4 for n ≥ 1. Prove that the sequence (xn)n≥1 isconvergent and calculate the limit limn→∞ xn

Define (xn)n≥1 by x1 := 1 and xn+1 := (3xn + 3)/4 for n ≥ 1. Prove thatlimn→∞ xn = 3.(Hint: First show that xn ≤ 3 for all n ≥ 1.

Construct two divergent sequences (x′n)n≥1 and (y′n)n≥1 such that (x′ny′n)n≥1 converges

1/1

Upgrade your grade with Knowee

Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.