Show that $S_n$ can be generated from the two-cycles $(1\,2),(1\,3),\dotsc,(1\,n)$.
I have a lemma given:
Lemma: Every $\sigma \in S_n$ can be written as a product of transpositions.
So I concluded that all I need to show is that $(1\,2),(1\,3),\dotsc,(1\,n)$ can generate all transpositions. Now, we have
$$(i\, j) = (1\,i)*(1\,j)*(1\,i)$$
for any $i, j \in \{2,\dotsc,n\}$ implying that we can generate all transpositions from the given two-cycles. Then I apply the lemma to conclude that every $\sigma \in S_n$ can be generated from the two-cycles. Please give me some feedback on my proof please!