SUPPOSE versus IF

Question

What is the difference between
suppose . . ., then . . .
and
if . . ., then . . .
?

For instance, in this...
"Let P(n) be a statement that is defined for all nEZ and let a be a fixed integer. Suppose that both of the following statements are true:

1. P(a) is true
2. for every integer, k ≥ a, if p(k) is true then p(k+1) is true.

Then the statement: "for all integers n ≥ a, p(n)" is true."

Why is it suppose that as opposed to if

Answer 1

As single sentences, "Suppose X, then Y" and "If X, then Y" are synonymous.

We say "Suppose X." when we want to keep X around as an assumption for the next while. This might just be because we have a statement of a theorem that's too long to fit into a single sentence. That's all that is going on in your example.

In proofs, we might prove an implication of the form "If X, then Y" by saying "Suppose X", continuing with several steps which are free to use X as an assumption, and concluding Y.