Using Jensen inequality to prove n / (1 / A1 + 1 / A2 +...) +1/an

Using Jensen inequality to prove n / (1 / A1 + 1 / A2 +...) +1/an

The conditions are A1, A2,..., an > 0
From the logarithmic function ln (x) is convex, using Jensen's inequality
(ln(a1)+ln(a2)+...+ln(an))/n

A high school mathematical inequality proves that a1 + A2 + ···· + an ≥ n * (the open n of A1 * A2 * ···· * an
A high school mathematical inequality proves that a1 + A2 + ···· + an ≥ n * (the open n power of A1 * A2 * ···· an) I'm not very smart
If I have information, I won't ask

There are many methods to prove the mean inequality, here is one;
It is obviously true when n = 1,2,
Suppose n = K (K ≥ 2),
When n = K + 1, if A1 = A2 = =a(k+1),
The formula naturally holds,
When A1 If two of a (K + 1) are not equal,
Let A2 ≤ A1 ≤ ≤a(k+1),
Let P = A1 × A2 ×····× a (K + 1) to the open (K + 1) power,
Then A1 × ×a(k+1)=p^(k+1),
And A1 < p

Factorization of (a + 1) 2 + A2 + (A2 + a) 2
2 is square