How does proof by induction work?

When using proof by induction we most often prove a statement P for positive integers n. We think about the problem in a domino-toppling fashion. The first step is to write out P(n=1), so inserting 1 for n in P. Showing that the left hand side LHS equals the right hand side RHS will prove P for n=1, so P(1). In the second step we assume that P is true for some positive integer k, so P(n=k). We write k instead of n into LHS and RHS. In the third step we evaluate P(n=k+1), so we plug in k+1 to the LHS, and use step 2 to rearrange the expression. Through rearranging we show that LHS of P(n=k+1) equals the RHS expected when plugging in k+1 into P. These 3 parts connect to form a proof for all n. Since we can say k=1, and we showed P(k+1) is true if P(k) is true (and we know P(k)=P(1) is true), we can say P(2) is true. Then repeating this mechanism means that P(3), P(4), P(5), ... are all true, thereby proving P(n) for all positive integers n.