How do we show $P(A) leq P(A Delta B) + P(A cap B) leq P(ADelta B) + P(B).$?

Based on your comments you meant $lvert{P(A)-P(B)}rvert leq P(ADelta B)$

$A subseteq (A Delta B) cup (A cap B)$, and the two sets on the RHS are disjoint. So $P(A) leq P(A Delta B) + P(A cap B) leq P(ADelta B) + P(B).$

So $P(A) - P(B) leq P(ADelta B)$. We can similarly show that $P(B)-P(A) leq P(ADelta B)$ and the result follows.