Prove $k+1>k $ where $k$ is an integer I'm being asked to prove that if $k∈ \Bbb Z$, $k+1>k$. Judging from our instructions, it appears (I am unsure) as though I cannot use the law of induction to solve this.  A hint gives that the proof depends on $1∈\Bbb N$. I was thinking of approaching this by using if $k∈\Bbb N$ then $x+k∈\Bbb N$.  Problem is, we were not given $k∈\Bbb N.$  Is this solvable without induction?
We are given associativity, commutivity, and the identity elements for addition and multiplication, the additive inverse, the properties of $=$, and basic set theory and logic.
Order for the integers is given by Let $m,n,p∈\Bbb Z$.  If $m<n$ and $n<p$, $m<p$.
Also assumed, and possibly relevant is if $x∈\Bbb Z$, then $x∈\Bbb N$ or $-x∈ \Bbb N$ or $x=0$.  

If you assume that $1 > 0$ is true, so it's trivial because, given $k \in \mathbb{Z}$,
$$
\mbox{ the inequation } \quad k + 1 > k \qquad \mbox{ is equivalent to } \qquad 1 > 0
$$
In general, if $x , y , z \in \mathbb{R}$,
$$
\mbox{ the inequation } \quad x > y \qquad \mbox{ is equivalent to } \qquad x + z > y + z
$$
and you can replace "$>$" by "$\geq$", "$<$", "$\leq$" or "$=$".
