Without loss of generality assume that every projection is bounded with a constant K_i>0, which means that for all x=(x_1,x_2,...)

|| x_i e_i || <= K_i || x ||

where e_i is the i-th unit vector.

Then in particular for the vector x, defined by x_i:=i*K_i/|| e_i ||

|| x_i e_i ||=i*K_i <= K_i || x ||

=> || x || >= i

for all i

which is impossible.