I'm reading Differential Forms in Algebraic Topology from Bott,Tu and i confused in section of spectral sequence of filtered complexes. In this section book consider ungraded complexes They're just forgetting about graduation in a complex. Given $K$ with differencial $D$ and a filtration $K=K_0\supset K_1\supset K_2\supset K_3\supset...$ we define $A=\bigoplus_{p\geq0}K_p$ which is a differential complex consider $D$ acting in each coordinate (at least that's what i understand). Then we define $i:A\to A$ as the sum of inclusions $K_{p+1}\hookrightarrow K_p$ and obtain exact sequence $$0\to\bigoplus_{p\geq0}K_p\overset{i}{\to}\bigoplus_{p\geq0}K_p\to \bigoplus_{p\geq0}K_p/K_{p+1}$$
Now, in the grading case the previous sequence gives an exact sequence of differential complexes and then we have a long exact sequence between cohomology groups
$$...\to H^k\left(\bigoplus_{p\geq0}K_p\right)\to H^k\left(\bigoplus_{p\geq0}K_p\right)\to H^k\left(\bigoplus_{p\geq0}K_p/K_{p+1}\right)\to H^{k+1}\left(\bigoplus_{p\geq0}K_p\right)\to...$$
and then sum all over cohomology groups we obtain an exact couple. I would appreciate it if someone could confirm if what I am saying is correct and if they could explain the following to me. Following the above (page 158) the book says " It is not difficult to see that the same diagram exists in the ungraded case". What would the sequence in cohomology look like in the ungraded case? And then the book does an example with a finite filtration
$$K=K_0\supset K_1\supset K_2\supset K_3\supset\{0\}$$
but i don´t understand how obtain the exact couple in this example. Please someone explain it to me. Thanks.