What kind of root? -th root, where is a positive integer?

Many things are wrong with this. First, a finite set can't be put in bijection with an infinite set! Two sets can be put in bijection with each other only if they have the same cardinality.Another question was, Let S be a finite set where z=a+bi. Prove S is bounded.

For this one, I said since S is finite, S is can be put in a 1-1 correspondence with .

Also, I was allowed to assume .

I then said .

Now, we have .

So S must be bounded. I know it is wrong but how should it be done or what could I have added or altered to make it correct?

Second, even supposing you had established a bijection, that doesn't mean ! I can put my fingers in bijection with , but that doesn't mean my fingers are themselves integers between 1 and 10.

Finally, even if were a subset of , that wouldn't mean that it's bounded. For instance itself isn't bounded (as a subset of ).

What you should have said is that the set consisting of the moduli of the elements of is a finite set of real numbers. A finite set of real numbers always has a finite upper bound - you should be able to prove this! This upper bound is a number such that for every , i.e. it's a bound for as a subset of .