I have a small question on Tate's proof of the structure of the group $K_2\mathbb Q$, as found in e.g., Milnor's book "Introduction to Algebraic $K$-theory".
The proof goes by showing that the map $K_2\mathbb Q\to\bigoplus_{p\text{ prime}}A_p$ (where $A_2=\mathbb Z/2$ and $A_p=(\mathbb Z/p)^\times$ for $p>2$) induced by the $p$-adic symbols is an isomorphism. To this end, one defines subgroups $L_m$ of $K_2\mathbb Q$ generated by elements $\{x,y\}$ for $x$, $y$ integers of absolute value $\le m$, and shows that for a prime $p>2$, $L_p/L_{p-1}\cong A_p$, and $L_1=L_2\cong A_2$. Then, by induction one has $L_p\cong \bigoplus_{2\le q\le p} A_q$ and the statement follows by passing to the colimit.
Now I just have a small question about the induction step. One assumes $L_{p-1}\cong\bigoplus_{2\le q\le\ell}A_q$ ($\ell $ being the greatest prime $\le p-1$) and knows $L_p/L_{p-1}\cong A_p$, but why is it obvious that the desired isomorphism follows from this?