In this proof of the Lemma 2.34 of Algebraic Topology I don't understand 2 things.
$(1)$ Hatcher defines $Y_i:=T\cup (X\times[i,\infty])$, then I think that it's obvius the fact that $Y_i$ deformation retracts onto $Y_{i+1}$, since $[i,\infty]$ deformation retracts onto $[i+1,\infty]$.
$(2)$ With the retractions of $Y_i$ in $Y_{i+1}$ how I can obtain a deformation retract of $X\times [0,\infty)$ onto $T$?
Concerning (1), we cannot just expect a deformation retraction of $X\times \lbrack i,\infty )$ to $X\times \lbrack i+1,\infty )$ to extend to a deformation retraction of the larger space $T\cup X\times\lbrack i,\infty )$. For instance, $S^1$ is the union of its northern and southern (closed) hemispheres. Both hemispheres deformation retract to a point, but there is no deformation retraction of $T$ which, e.g., is the identity on the southern hemisphere while deforming the northern hemisphere to a point.
As for (2), the point is to compress each $Y_i$ in increasingly short time intervals. The resulting homotopy is well-defined and continuous, essentially due to the "pasting lemma" from elementary point-set topology; the intervals $\lbrack 1 - 1/2^i, 1 - 1/2^{i+1}\rbrack$ form a closed cover of the unit interval and the partial homotopies agree on the intersections $\lbrace 1/2^i\rbrace$ by construction. This allows us to eventually compress every $Y_i$ (this is no weirder than Achilles catching up to the turtle).