Real Limit Algebra

Why

Consider two convergent sequences. What if we add them termwise? Or multiply?

Let

(x_{n})_{n \in N}

and

(y_{n})_{n \in N}

be two limits with

x_{0}

and

y_{0}

R

. Then the sequence

(s_{n})

defined by

s_{n} = x_{n} + y_{n}

converges to the limit

x_{0} + y_{0}

and the sequence

m_{n} = x_{n} y_{n}

converges to the limit

x_{0} y_{0}

.¹

In particular for $a \in R$ , the sequence $(c_{n})$ defined by $c_{n} = a x_{n}$ converges to the limit $a x_{0}$ .