Prove $|A| = |B|$

Question

Let $A= \{a_1,a_2,a_3,\ldots\}$. Define $B = A − \{a_{n^2} : n \in\mathbb N\}$. Prove that $|A|=|B|$.

I would say that $B = \{a_2,a_3,a_5,a_6,\ldots\}$. Thus $B$ is a infinite subset of $A$ and since $A$ is denumerable, $B$ is also denumerable. I do not know how to proceed from here.

Answer 1

You pretty much finished. Just remember that $X$ is denumerable if and only if $|X|=|\Bbb N|$, and that equicardinality is an equivalence relation.

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Another option is to show there is an injection from $A$ into $B$. Then use the Cantor-Bernstein theorem.

Answer 2

Without further conditions the statement is wrong as Dustan has already pointed out in a comment. Take for example $a_i=0$ for all $i$, then $|A|=1$, $|B|=0$.