We have S defined as: S = {x|x in R, x>=0, x^2< c}

How do I show that c+1 is an upper bound for S, and that S has a least upper bound denoted by b??

I tried doing it this way:

If c<1, we have x<1<1+c, i.e, 1+c is an upper bound for S

If c>1, then x<sqrt(c)<c<c+1 and thus, c+1 is an upper bound for S.

I am still unsure if this is the correct method!! Any suggestions?