How to prove without using equipotence of sets this property of cardinal numbers?

Question

Let $\aleph _{0}$ the cardinal of the natural numbers. If $b$ is infinite then $\aleph _{0}+b=b$

I don't know how to start, i've tryed doing this:

if $b$ is an infinite cardinal number then $\aleph _{0} \leq b$ then exist a cardinal number $p$ such that $\aleph _0 +p=b$. But i don't know how to get that $p=b$.

I aprecciate yout help.

Answer 1

[It's really not clear to me what tools you're allowed to use here given that you're supposed to do this "without using equipotence of sets", but here's the closest thing to that I can see.]

Hint: Since $\aleph_0\leq b$, there exists a cardinal $c$ such that $b=\aleph_0+c$.

A full proof is hidden below:

Let $X$ be a set of cardinality $b$. Since $\aleph_0\leq b$, there exists an injection $f:\mathbb{N}\to X$. Let $c$ be the cardinality of $X\setminus f(\mathbb{N})$. Then $b=\aleph_0+c$. Now just observe that $$\aleph_0+b=\aleph_0+(\aleph_0+c)=(\aleph_0+\aleph_0)+c=\aleph_0+c=b.$$