Prove that $\sum_{i=1}^{n} |\sin a_i| +|\cos s_n|\ge 1$

Question

Let $a_i$ be a real sequence and $s_n=\sum_{i=1}^{n}a_i$ its sequence of partial sums. Prove that $$\left(\sum_{i=1}^{n} |\sin a_i|\right) +| \cos s_n|\ge 1$$

I have a proof by induction.

Let us take the base case $n=1$:

$$|\sin a_1|+|\cos a_1|\ge |\sin^2a_1|+|\cos^2a_1|=1$$

Assuming for n=k is true, it is sufficient to prove

$$|\sin a_{k+1}|+|\cos s_{k+1}|\ge |\cos s_k|,$$

which is obvious using $|\cos s_k|=|\cos(s_{k+1}-a_{k+1})|$

However, I am looking for a proof without using induction.

Any hints??

Answer 1

Notice that $\sum|\sin a_i|\ge\left|\sin\sum a_i\right|=|\sin s_n|$ from the Triangle Inequality and we know that $|\cos s_n|+|\sin s_n|\ge1$.