## Let S be as set of real numbers , and let \${x_n}\$ be a sequence which converges to l

Let S be as set of real numbers , and let \${x_n}\$ be a sequence which converges to l. Suppose that for every \$n inmathbb{N},x_n\$ is an upper bound for S . prove l is an upper bound of S And also …

## \$mathbb{N}\$ complete metric space

I’m trying to prove that \$(mathbb{N},d)\$ is a complete space where \$d=left | m-n right |\$. So I define \$a_{n}:=n\$ , if it’s cauchy we know that \$forall epsilon\$>0 there exist \$Nin mathbb{N}\$ s….

## How can we have more than one sequence in a set?

I am looking at a proof for a divergence criterion for functional limits: Let \$f\$ be a function defined on \$A\$, and let \$c\$ be a limit point of \$A\$. If there exist two sequences \$(x_n)\$ and \$(y_n)\$ …

## Find \$sum_{n=0}^inftyfrac1{2^ncdot n!}\$ and \$sum_{n=0}^inftyfrac1{4n+3}\$

Consider the set of integers \$S = {2^0.0!, 2^1.1!, 2^2.2!, ldots }\$. What do we get when we sum its series of reciprocals? Answer the same question for set \$S’ ={3, 7, 11, 15, 19, ldots }\$. I …

## Limiting behaviour of \$sum_{j=0}^infty|sum_{k=1}^na_{j+k}|^p\$

I would like to prove or disprove the following statement. Suppose that \${a_j:jge0}\$ is an absolutely summable real sequence, i.e. \$sum_{j=0}^infty|a_j|

## I need help using the limit comparison test for \$sum frac{1}{sqrt{n^2 + 1}}\$

I need to determine whether the following series converges or diverges: \$\$sum_{n=1}^{infty} frac{1}{sqrt{n^2 + 1}}\$\$ I’m having trouble finding a series to compare this to but I was thinking …