Prove if A is a set finite and B is infinite then $|A|+|B|=|B|$

Question

Prove if $A$ is a finite set and $B$ is infinite then $|A|+|B|=|B|$

My work

I was thinking in solve this exercise by contradiction.

Suppose $|A|+|B|\not = |B|$ then $|A|+|B|=|A|$

Here im a little stuck. Can someone help me?

Answer 1

Hint: First show that $|B|+1=|B|$, then use induction over the size of $A$.

Answer 2

Hint: adding a single element to a countably infinite set yields a set of the same cardinality...

Consider $\mathbb N$. Define $\phi:\mathbb N\cup \{x\}\to\mathbb N$ by $\phi (x)=1$ and $\phi (n)=n+1 \, \forall n\in\mathbb N$. $\phi$ is a bijection.

As Cantor said, "I see it; but I can't believe it."