Question

If (x_n) and (y_n) are sequences in the same normed space X, show that x_n→x and y_n→y implies x_n+y_n→x+y as well as αx_n→αx, where α is any scalar.

Answer 1

The set of all sequences in the normed space X is a vector space under the natural component-wise definitions of vector addition and scalar multiplication:

$\{x_1,x_2,...\}+\{y_1,y_2,...\}=\{x_1+y_1,x_2+y_2,...\}$ and

$\alpha\{x_1,x_2,...\}=\{\alpha x_1,\alpha x_2,...\}$

So, if  x_n→x and y_n→y, then:

x_n+y_n→x+y and αx_n→αx, where α is any scalar.