I have a function:
$$\pi_i^1=y-g_i+a\sum^n_{j=1}g_j,$$
where 0 < a<1< na,
and I need to prove this: $$\frac{\partial(\sum^n_{i=1}\pi^1_i)}{\partial g_i}=-1+na>0.$$
I am not very experienced with summations to be confident in doing this.
This is what I have done so far (below), but I am not sure it is correct. Can anyone help? $$\sum^n_{i=1}\pi_i=\sum^n_{i=1}y-\sum^n_{i=1}g_i+\sum^n_{i=1}\sum^n_{j=1}ag_j\\ =ny-\left(g_i+\sum^n_{j\ne i}g_j\right)+a\sum^n_{j=1}ng_j\\ =ny-\left(g_i+\sum^n_{j\ne i}g_j\right)+na\sum^n_{j=1}g_j\\ =ny-\left(g_i+\sum^n_{j\ne i}g_j\right)+na(g_i+\sum^n_{j\ne i}g_j)$$