Although the following does not provide another proof (perhaps it is possible to attempt one on this basis) I found it nice to see the following pictures.         
Let's take from the series $f(x) = \sum_{k=0}^\infty {x^k \over \sqrt{k!}}$ the following variants in the same spirit as we have the hyperbolic and trigonometric series from the exponential-series:
$$\begin{array}{}
 \small \exp_{\tiny \sqrt{\,}}(x) &=& f(x) \\
 \small \cosh_{\tiny \sqrt{\,}}(x) &=& \sum_{k=0}^\infty {x^{2k} \over \sqrt{(2k)!}} \\
 \small \sinh_{\tiny \sqrt{\,}}(x) &=& \sum_{k=0}^\infty {x^{2k+1} \over \sqrt{(2k+1)!}} \\
\small  \tanh_{\tiny \sqrt{\,}}(x) &=& { \sinh_{\tiny \sqrt{\,}}(x)\over \cosh_{\tiny \sqrt{\,}}(x)  } \\
\small  \cos_{\tiny \sqrt{\,}}(x) &=& \sum_{k=0}^\infty (-1)^k {x^{2k} \over \sqrt{(2k)!}} \\
\small  \sin_{\tiny \sqrt{\,}}(x) &=& \sum_{k=0}^\infty (-1)^k {x^{2k+1} \over \sqrt{(2k+1)!}} \\
\end{array}$$ 
The answer to your question is equivalent to say, that always (="for real $x$")          

 - $\small \cosh_{\tiny \sqrt{\,}}(x)$ is larger than $\small \sinh_{\tiny \sqrt{\,}}(x) $ $\qquad \qquad$ or that
 - $\small \mid \tanh_{\tiny \sqrt{\,}}(x) \mid \lt 1$

<hr>

To illustrate this I've plotted the $\sinh_{\tiny \sqrt{\,}}$ and $\cosh_{\tiny \sqrt{\,}}$-curves:              

![bild1][1]      

This gives surely an extremely familiar impression...            


The $\tanh_{\tiny \sqrt{\,}}$-curve looks completely familiar too:
![bild2][2]                  

and the image suggests, that indeed the absolute value of $\small  \tanh_{\tiny \sqrt{\,}}(x) $ very likely is smaller than $1$ for all real $x$.


<hr>
However, things are different for the $\sin_{\tiny \sqrt{\,}}$ and $\cos_{\tiny \sqrt{\,}}$ curves - they deviate strongly from the nicely periodic common trigonometric functions:                  

![bild3][3]                

and combined they do not give a circle, but some ugly thing, strongly distorted (y-axis by $\small \cos_{\tiny \sqrt{ \,} }(\phi)$, x-axis by $\small \sin_{\tiny \sqrt{ \,} }(\phi)$, $\phi$ from $-5$ to $+5$) :             

![bild4][4]

<hr>
  [1]: https://i.sstatic.net/zKZ6I.png
  [2]: https://i.sstatic.net/DT4At.png
  [3]: https://i.sstatic.net/yiYpF.png
  [4]: https://i.sstatic.net/YZSCk.png