Answer
See explanation
Work Step by Step
$k=1$
Any $1 \times 1$ matrix is lower triangular, and product of two $1 \times 1$ matrices is $1 \times 1$ matrix, so result is also lower triangular.
Let that for $n=k,$ product of two $k \times k$ lower triangular matrices is lower triangular.
Make two $(k+1) \times(k+1)$ matrices $A$ and $B$
Partition these matrices as:
$A=\left[\begin{array}{ll}a & 0^{T} \\ v & A_{1}\end{array}\right], B=\left[\begin{array}{ll}b & 0^{T} \\ w & B_{1}\end{array}\right],$ where $A_{1}$ and $B_{1}$ are $k \times k$ lower triangular
matrices (they are lower triangular because $A$ and $B$ are lower triangular).
Then:
$A B=\left[\begin{array}{ll}a & 0^{T} \\ v & A_{1}\end{array}\right] \cdot\left[\begin{array}{ll}b & 0^{T} \\ w & B_{1}\end{array}\right]=\left[\begin{array}{cc}a b+0^{T} w & a 0^{T}+0^{T} B \\ v b+A w & v 0^{T}+A B\end{array}\right]=\left[\begin{array}{c}a b \\ b v+A w & A B\end{array}\right]$
This shows that the product is also lower triangluar by principle of induction.
