Proof of DCF model by mathematical induction
I suspect your induction hypothesis should be an expression for $P_t$ not $P_{t+j}$ , and should be discounting $P_{t+j}$ not $s$ times but $j$ times i.e. 

$$P_{t} = \left(\sum\limits_{s=1}^{j} \frac{D_{t+s}}{(1+k)^s}\right) + \frac{P_{t+j}}{(1+k)^j}$$

for $j \ge 1$, which reduces to your initial statement $P_t = \frac{D_{t+1}}{(1+k)} + \frac{P_{t+1}}{(1+k)}$ when $j=1$

Now consider your initial statement replacing $t$ by $t+j$ to give  $P_{t+j}= \frac{D_{t+j+1}}{(1+k)} + \frac{P_{t+j+1}}{(1+k)}$ and substitute this into your induction hypothesis to give 

$$P_{t} = \left(\sum\limits_{s=1}^{j+1} \frac{D_{t+s}}{(1+k)^s}\right) + \frac{P_{t+j+1}}{(1+k)^{j+1}}$$

which proves that the corrected induction hypothesis follows from your original statement