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If S[tex]\subset[/tex]R is finite and non-empty, then S has a maximum.

Can someone look over this? I struggled a bit in my first proof class, which is why I am asking for help, so I really am unsure if this is right at all.

Let S={1}

So 1[tex]\in[/tex]R such that for all x[tex]\in[/tex]S, 1[tex]\geq[/tex]x

So 1 is an upper bound for S

1[tex]\in[/tex]S, so by definition 1=max S

Let S={m+1}

Then m+1>m for all m[tex]\in[/tex]S

So m+1 is an upper bound for S

Since m+1[tex]\in[/tex]S, then by definition m+1=max S

Therefore if S[tex]\subset[/tex]R is finite and non-empty, then S has a maximum.

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# Intro Analysis - Proof - max M

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