Answer
See answer below
Work Step by Step
Assume that the set of vectors $\{v_1,v_2,...v_k$ is linearly independent (1)
and $v_{k+1} \notin$ span $\{v_1,v_2\}$ .
Let $c_1,c_2,...c_k, c_{k+1}$ be scalars
$\rightarrow c_1v_1+c_2v_2+...+c_kv_k+c_{k+1}v_{k+1}=0$
Suppose $c_{k+1} \ne 0 \rightarrow v_{k+1}=-\frac{c_1}{c_{k+1}}v_1-\frac{c_2}{c_{k+1}}v_2$ so $v_{k+1}$ will $in$ span $\{v_1,v_2,...v_{k}\}$
Hence, $c_{k+1}=0 \rightarrow c_1v_1+c_2v_2+....+c_kv_k+c_{k+1}v_{k+1}=c_1v_1+c_2v_2+...+c_kv_k+0=c_1v_1+c_2v_2+...+c_kv_k$
$\rightarrow c_1v_1+c_2v_2+...+c_kv_k=0$ (2)
From (1) and (2) $\rightarrow c_1=c_2=...=c_k=0$
$\rightarrow c_1=c_2=...=c_k=c_{k+1}=0$
Hence, the set $\{v_1,v_2,...v_k,v_{k+1}\}$ is linearly independent.
