1. Proof by induction

Prove by induction on $n$ that any set of $n$ reals is bounded.

So the base case for $n = 1$ is true because consider the set $\{a\}$ where $a \in \mathbb{R}$ then there exists a number $a+\epsilon$ where $\epsilon \in \mathbb{R}^+ \cup \{0\}$ which is an upper bound and a number $a-\epsilon$ which is a lower bound hence the set $\{a\}$ is bounded.

Now the inductive hypothesis is that assume any set of $k$ reals is bounded.

How do I proceed on to prove that any set of $k+1$ reals is bounded?

Thanks

2. Re: Proof by induction

The assumption is that $\{r_1,\dots,r_k\}$ is bounded, say by $M$ above and $-M$ below.

Now consider the set $S = \{r_1,\dots,r_k,r_{k+1}\}$. By assumption we have $|r_j| \leq M$, for $j = 1,\dots,k$.

Let $M' = \max\{M,|r_{k+1}|\}$. Show that for any $r_j \in S$, we have $|r_j| \leq M'$.