To prove that ( (a_n) ) approaches 0 when ( lim_{n to infty} frac{a_n}{a_{n+1}} > 1 ), we can use the ratio test.

Given ( lim_{n to infty} frac{a_n}{a_{n+1}} > 1 ), let’s assume ( L = lim_{n to infty} frac{a_n}{a_{n+1}} ). Since ( L > 1 ), there exists ( N ) such that for all ( n > N ), ( frac{a_n}{a_{n+1}} > 1 ).

By rearranging the inequality, we have ( a_n > a_{n+1} ) for all ( n > N ). This implies that the sequence ( (a_n) ) is strictly decreasing for ( n > N ).

Since ( (a_n) ) is strictly decreasing and bounded below, it converges to a limit ( L’ ). Let’s denote this limit as ( lim_{n to infty} a_n = L’ ).

Now, taking the limit as ( n to infty ) in the given expression ( frac{a_n}{a_{n+1}} ), we get ( L’ = lim_{n to infty} frac{a_n}{a_{n+1}} = L > 1 ).

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