Let $\displaystyle x$ be a point in $\displaystyle X$. My question is about the following short exact sequence:

$\displaystyle 0 \to H_1(X) \to H_1(X,\{x\}) \to H_0(\{x\}) \: \xrightarrow{i_\ast} \: H_0(X)\to H(X,\{x\})\to 0,$

where $\displaystyle i_* : H_0 (\{ x \} ) \to H_0(X)$ is the homomorphism induced by the obvious inclusion. I am told that $\displaystyle i_*$ is injective. Although this does make some sense intuitively, I can't really prove it. Can someone show me why it is?