I'm having trouble understanding the proof of Lemma 2.34 in Hatcher's "Algebraic Topology". More specifically I don't understand the proof of the following statement if $k=n-1$:
The inclusion $i:X^n\to X$ induces an isomorphism $i_*:H_k(X^n)\to H_k(X) \ \ \forall k<n$
If $k<n-1$ I understand that the extreme homologies of this segment vanish:
$$ {H_{k+1}(X^n,X^{n-1})} \rightarrow H_{k}(X^{n-1}) \rightarrow H_{k}(X^{n}) \rightarrow {H_{k}(X^n,X^{n-1})} $$
So the central map is an isomorphism and then inductively $i_*$ is an isomorphism. But if $k=n-1$, then: $$ \mathbb{Z}\text{Cells}_n \rightarrow H_{n-1}(X^{n-1}) \rightarrow H_{n-1}(X^{n}) \rightarrow0 $$
I understand that the central map is surjective(because of the $0$ on the right), but I don't see how the desired result follows from this.
In the highlighted statement that you are asking about, the goal is to understand an induced homology homomorphism of the inclusion $X^n \hookrightarrow X$. For that purpose, I don't think there's any relevance to the inclusion $X^{n-1} \hookrightarrow X$.
Instead you should be studying the sequence of inclusions $$X^n \hookrightarrow X^{n+1} \hookrightarrow X^{n+2} \hookrightarrow \cdots $$ Your question then suggests studying the induced homology homomorphisms of this sequence in dimension $n-1$: $$(*) \qquad H_{n-1}(X^n) \mapsto H_{n-1}(X^{n+1}) \mapsto H_{n-1}(X^{n+2}) \mapsto \cdots $$ I suspect that with what you already know, you can show that these homomorphisms are all isomorphisms. Just use the long exact homology sequence of the pair $(X^{i+1},X^i)$ for $i \ge n$, exactly as you have been doing, focussing on the portion $$H_n(X^{i+1},X^i) \mapsto H_{n-1}(X^i) \mapsto H_{n-1}(X^{i+1}) \mapsto H_{n-1}(X^{i+1},X^i) \quad\quad (i \ge n) $$ and noting that the outside terms are zero.
Once that's done, then by a direct limit argument applied to the sequence $(*)$, you'll conclude that $H_{n-1}(X^n) \to H_{n-1}(X)$ is an isomorphism.