- #1
bugatti79
- 794
- 1
If lim x_n=x n to infinity and lim y_n=y n to infinity
prove rigorously
lim n to infinity (x_n/5+10y_n)=x/5+10y.
My attempt
let ε>0. Must find [itex]n_0 \in \mathbb{N}[/itex] such that
[itex]||(x_n/5+10y_n)-(x/5+10y)||<ε[/itex] for all [itex]n>n_0[/itex]
[itex]||(x_n/5+10y_n)-(x/5+10y)||=||(x_n/5-x/5)||+||10y_n-10y|| \le ||(x_n/5-x/5||+||(10y_n-10y)||[/itex]
since [itex]x_n=x[/itex] for the limit n to infinity and similarly for y_n and given ε>0 then ε/2>0
so there exist [itex]n_1 \in N[/itex] such that [itex]||x_n/5-x/5||< ε/2[/itex] for all [itex]n \ge n_1[/itex]
and
[itex]||10y_n-10y||< ε/2[/itex] for all [itex]n \ge n_2[/itex]
Let [itex]n_0=max{n_1,n_2}[/itex], then for all [itex]n \ge n_0[/itex]
implies [itex]||(x_n/5+10y_n)||-||x/5+10y|| \le ||(x_n/5-x/5||+||(10y_n-10y)||<ε/2+ε/2=ε[/itex]...?
prove rigorously
lim n to infinity (x_n/5+10y_n)=x/5+10y.
My attempt
let ε>0. Must find [itex]n_0 \in \mathbb{N}[/itex] such that
[itex]||(x_n/5+10y_n)-(x/5+10y)||<ε[/itex] for all [itex]n>n_0[/itex]
[itex]||(x_n/5+10y_n)-(x/5+10y)||=||(x_n/5-x/5)||+||10y_n-10y|| \le ||(x_n/5-x/5||+||(10y_n-10y)||[/itex]
since [itex]x_n=x[/itex] for the limit n to infinity and similarly for y_n and given ε>0 then ε/2>0
so there exist [itex]n_1 \in N[/itex] such that [itex]||x_n/5-x/5||< ε/2[/itex] for all [itex]n \ge n_1[/itex]
and
[itex]||10y_n-10y||< ε/2[/itex] for all [itex]n \ge n_2[/itex]
Let [itex]n_0=max{n_1,n_2}[/itex], then for all [itex]n \ge n_0[/itex]
implies [itex]||(x_n/5+10y_n)||-||x/5+10y|| \le ||(x_n/5-x/5||+||(10y_n-10y)||<ε/2+ε/2=ε[/itex]...?