1. Surjectivity

Let f:A→B. Prove that f is surjective if and only if, for all sets C and functions g,h:B→C, g∘f = h∘f implies g = h.

2. A recursively-defined function

Let f:ℕ→ℤ be defined by the rule:

where a,b∈ℤ.

Show that there exist constants c and d (which may depend on a and b but not on n), such that f(n) = cn+d for all n∈ℕ.

Prove that the following identity holds for all n∈ℕ:

$1 + \sum_{k=1}^{n} \left(k \cdot \prod_{i=1}^{k} i\right) = \prod_{i=1}^{n+1} i.$