xn ⊆ R2. Let xn=(x1n, x2n)→(x1, x2) (meaning by coordinate convergence).
Now, Axn=(ax1n+bx2n, cx1n+cx2n) is also a vector and we need to show that both coordinates converge.
Since x1n and x2n converge and a, b are real numbers, then
ax1n+bx2n →(ax1+bx2). The same with (cx1n+dx2n).
Axn=(ax1n+bx2n, cx1n+cx2n)→(ax1+bx2, cx1+dx2)=Ax