- #1

CuppoJava

- 24

- 0

I devised this problem for myself as part of a bigger question that I'm working on, and am having trouble solving it. I think it involves a nested induction proof but I am not sure how to start. A tip on how to begin would be much appreciated.

Thanks

-Patrick

## Homework Statement

[itex]

S_0 = \{\}

[/itex]

[itex]

S_i = \{0\} \bigcup \{s + \pi | s \in S_{i-1}\}

[/itex]

Show that [itex] S_i \subseteq S_{i+1} [/itex] for all [itex]i[/itex].

## The Attempt at a Solution

I started by converting the problem to the following form:

[itex]\forall i > 0, 0 \in S_i[/itex]

[itex]\forall s \in S_i, s + \pi \in S_{i + 1}[/itex]

Show that [itex] \forall s \in S_i, s \in S_{i + 1} [/itex]

But I don't know what to do next.