Let $X$ be a topological space and $CX$ be its corresponding cone. Suppose $g$ is a $n-$ cycle of $X$, I want to show that $g$ is a boundary in $CX$.
According to the naturality of the cone construction, I can find a map $Cg: \Delta^{n+1} \to CX$, intuitively the boundary of $Cg$ should be $g$. How to calculate the boundary of $Cg$ formally?
First, to set conventions: $$CX = I\times X/\{1\}\times X$$ $$X \cong \{0\}\times X \subseteq CX$$ $$\Delta^k = \{(t_o,\ldots,t_k) \mid \sum t_i = 1\}\subseteq I^{k+1}$$ We define the vertices $e_m \in \Delta^k$ by $$e_m = (t_0,\ldots,t_k), t_i = 0 \text{ for } i\not = m \text{ and } t_m = 1$$ Now, we identify subsimplices of $\Delta^k$ by listing the $e_m$ in some order, for example $[e_0,\ldots, \hat{e_i},\ldots, e_k]$ is the simplex identified with $\Delta^{k-1}$ by omitting the vertex $e_i$. Of course, omitting no vertices, $[e_0,\ldots, e_k] = \Delta^k$. To avoid excessive indexing, I will abuse notation and use the letters $e_i$ for the standard vertices of any dimensional simplex. It should be clear from the context which dimension we are in.
Now, we identify $C\Delta^k\cong \Delta^{k+1}$ via a map: $$(s,e_i) \mapsto se_0 + (1-s)e_{i+1}$$ In this way, the cone point corresponds to $e_0$. (This map is a homeomorphism by the standard compactness/Hausdorffness argument.)
Now, we define a natural transformation $C: S_*(X) \to S_{*+1}(CX)$ (singular chain functor) by sending a singular simplex $g:\Delta^k \to X$ to $Cg: \Delta^{k+1} \to CX$ (abuse of notation using the identification above). If you compute $\partial Cg$, then by definition you have $$\partial Cg = \sum_{i=0}^{k+1} (-1)^i (Cg)\vert_{[e_0,\ldots,\hat{e_i},\ldots, e_{k+1}]}$$ Now using the above identifications, it should be clear that $$(Cg)\vert_{[e_0,\ldots,\hat{e_i},\ldots, e_{k+1}]} = C(g\vert_{[e_0,\ldots,\hat{e_{i-1}},\ldots, e_{k}]})$$ for $i>0$, and, $$(Cg)\vert_{[\hat{e_0},\ldots, e_{k+1}]} = g$$ using the identifications above. Thus, $\partial Cg = g - C(\partial g)$, in other words, $C$ is a (natural) nullhomotopy of the map induced by the inclusion $X\to CX$ on singular chains. It follows immediately that this map sends cycles to boundaries.