The question is about equation in green color for No-Ponzi constraint. In the answer it said a(T+1) should > 0, I am not sure why it is not aT >0. I am also not clear where low bound equation comes from.

Consider an infinitely lived individual that maximizes the present value of expected lifetime utility max E 'u (el) {9.de+1}=0 subject to the budget constraint 4 +at+1 = (1+r) at + yt for all t = 0,1,2,..., (1) and a No-Ponzi constraint, lim Eo [at] > 0, (2) + (1+r) with r, and ao given, and {yt} following some exogenous stochastic process. Suppose that B(1+ r) = 1 and the utility function is given by u(at) = C1 Zbc, b>0. (a) Explain what the No-Ponzi constraint (2) means. What would be the solution to the problem if this constraint was not imposed? Would this constraint be needed if the time horizon was finite? Answer It means that the present value of the individual asset position in the long-run must be positive. Without this constraint the solution is trivial: drive 0t+1 towards - and consume every period, a pretty extreme Ponzi scheme. With a finite time horizon, say T, the constraint should be replaced by the constraint that individuals are not allowed to leave any debt, that is 07+1 > 0. More precisely, any lower bound, say ar+t > a, would rule out Ponzi schemes, but it would be optimal to leave any amount of debt that was allowed, that is, to set at +1 = a