Proof:

We use the additive notation and shall use the same symbol 0 to denote the identity element of all the groups. The cartesian product $\prod G_{\alpha}$ is a group with respect to component-wise addition and we consider the subgroup $\bigoplus G_{\alpha}$ given by

$$\bigoplus_{\alpha\in \Lambda}G_{\alpha} = \Big\{ (x_{\alpha})_{\alpha}\in \prod_{\alpha\in \Lambda}G_{\alpha}\;/\; x_{\alpha} = 0$$    for all but finitely many indices $$\alpha\; \Big\}.$$

For each $\beta \in \Lambda$ we define the standard inclusion map

$$\iota_{\beta} : G_{\beta}\longrightarrow \bigoplus_{\alpha\in \Lambda} G_{\alpha}$$

such that $\iota_{\beta}(x)$ has entry $x$ in position $\beta$ and all other coordinates are zero. We leave it to the reader to check that the group $\bigoplus_{\alpha}G_{\alpha}$ together with the family $\{\iota_{\alpha}:G_{\alpha}\longrightarrow G\;/\;\alpha \in \Lambda\}$ satisfies all the requirements.